Jake Gyllenhaal could do absolutely anything next. Asked over lunch about his reading of a famed line in “Othello” — Iago’s seething “I hate the Moor” — the 44-year-old actor furrows his brow, runs his hands through his buzzed hair and ponders the performance I attended. “I’m trying to think about yesterday’s matinee, because it changes,” he says. “I’ve not made any sort of definitive choice.”

Soft-spoken and gently smiling, Gyllenhaal later summons the wide-eyed mania of his zanier characters — think his drunken “Okja” zoologist or Mr. Music of “John Mulaney & the Sack Lunch Bunch” — when the topic shifts to his eclectic body of work. “I’m just kind of random,” the Oscar nominee exclaims. “I guess as focused and intense as I can be, I also have a sense of, like, ‘That sounds fun. Oh, that scares me — I’ll give that a shot.’”

Then there’s the matter of what I should order at Via Carota, the trendy West Village osteria Gyllenhaal picked for our mid-March meetup. Having narrowed my options to the cacio e pepe and the lemon risotto, I ask Gyllenhaal for his recommendation. “You want to get both?” he gleefully responds. “You’ve got to do it. Do both. Get both!” In the spirit of impulsivity, I embrace the idea. “I mean, you’re working — you should have some joy,” Gyllenhaal says. “We have to leave you carbed up. You have a lot of typing to do.”

That carpe diem approach helped steer Gyllenhaal toward “Othello,” the blockbuster Shakespeare revival now on Broadway. Gyllenhaal was shooting “Guy Ritchie’s The Covenant” on the Spanish island of Tenerife when he fielded the offer to star as Iago opposite Denzel Washington’s titular general in the Kenny Leon-directed tragedy. A Shakespeare novice, Gyllenhaal asked to read the play and promptly bumped against Iago’s first monologue.

“I read it through twice, and I went, ‘I don’t know,’” Gyllenhaal recalls. “There were bits I understood, and I sat in this purgatory of, ‘Can I do this?’”

Still, Gyllenhaal thought about how his and Washington’s paths had run parallel for years. Gyllenhaal worked with director Antoine Fuqua, Washington’s frequent collaborator, on the films “Southpaw” and “The Guilty.” When prepping for “Southpaw’s” boxing sequences, Gyllenhaal connected with Terry Claybon, Washington’s longtime trainer, and worked out at the same gym as his fellow A-lister.

Having floated in Washington’s orbit without colliding, Gyllenhaal was eager to connect with an acting icon he had revered for decades. A pre-pandemic New York theater regular, Gyllenhaal also hadn’t starred in a play since his Tony-nominated turn in 2019’s “Sea Wall/A Life” and found himself itching to fine-tune his technique onstage.

“Am I going to forever say Kenny Leon and Denzel Washington asked me to play Iago and I said, ‘Thank you, but no’?” Gyllenhaal asks. “Also, I think it was at a time [when I was] just finding a moment to be reinspired by what it is that I do.”

Thus Gyllenhaal signed on for the production, which runs through June 8 at the Ethel Barrymore Theatre, and spent the better part of a year becoming fluent in Shakespeare. As he studied with Columbia professor James Shapiro and vocal coach Jeannette Nelson, Gyllenhaal navigated the byzantine text and found a way into Iago’s troubled headspace. That part is nothing new — whether he’s playing a hallucinating teen in “Donnie Darko,” a tragically repressed sheepherder in “Brokeback Mountain” or a ruthlessly opportunistic videographer in “Nightcrawler,” Gyllenhaal has a knack for shining light on the mind’s darkest recesses.

As much as “fearless” gets thrown around acting circles with abandon, Gyllenhaal earns the moniker. Picking projects that traverse in torment, he’s not afraid to bring a character’s inner demons to the surface. Brawny-to-scrawny transformations, vocal affectations, physical tics and trembles — Gyllenhaal stuns by fusing intense preparation with in-the-moment inspiration.

“There’s an ambition to be at his best and to push the boundaries of acting, and to try to create something that has not been seen before,” says Denis Villeneuve, who directed Gyllenhaal in the surrealist drama “Enemy” and the child-abduction thriller “Prisoners.” “He wants to try to re-create the chaos of life.”

Speaking between sips of mint tea, Gyllenhaal is not so chaotic. Mostly, he’s cordial and considered, with dashes of self-deprecation to offset the earnestness. Clocking my empty plate, Gyllenhaal serves me a helping of insalata verde between name-checking Danny Kaye, Paul Newman and Washington as his acting idols and hailing the performances of Cole Escola in “Oh, Mary!” and Audra McDonald in “Gypsy” this Broadway season.

The son of accomplished screenwriter Naomi Foner Gyllenhaal and prolific TV director Stephen Gyllenhaal, he grew up in Los Angeles’ Hancock Park neighborhood immersed in show business. As a child watching his now-Oscar-nominated older sister, Maggie, perform onstage, he instinctively wanted to emulate her. After appearing in a pair of student productions when he was 10 or 11 — playing the Pharaoh in “Joseph and the Amazing Technicolor Dreamcoat” and the Scarecrow in “The Wizard of Oz” — Gyllenhaal discovered comfort in the theater.

“The oddity, I think, of performance is that it’s so unnatural,” Gyllenhaal observes. “But I remember something feeling right about that.”

As Gyllenhaal found his way to Hollywood — making his screen debut as Billy Crystal’s son in “City Slickers” at age 10 and breaking out as NASA engineer Homer Hickam in “October Sky” at 18 — he leaned on a lesson instilled by his industrious parents: “Freedom is on the other side of discipline.” Looking back, Gyllenhaal credits his filmmaker father with gifting him a sense of cinematic wonder and his wordsmith mother with fostering his appreciation for storytelling.

“They gave me a world — I think some of it was conscious and some was unconscious — saying, ‘Look at this sacred space where you can put up all these feelings and let them out,’” Gyllenhaal says.

Actor Peter Sarsgaard had just started dating Maggie Gyllenhaal when he met Jake at a bar in the East Village more than two decades ago. Gyllenhaal naturally had questions — Sarsgaard was the guy seeing his sister, after all. Yet over the years, Gyllenhaal’s now-brother-in-law has seen that inquisitive streak endure.

“If he finds something there, then his curiosity won’t let up,” says Sarsgaard, who went on to work with Gyllenhaal on the 2005 Gulf War drama “Jarhead” and the 2024 courtroom drama series “Presumed Innocent,” among other projects. “Many good actors are people that are fervent. Onstage, you’re going back over the same thing over and over, and you have to be curious in order not to be bored out of your mind. On film, it just goes on for ages. So it really requires an unrelenting attitude.”

Boasting both leading man charm and character actor eccentricity, Gyllenhaal has often gravitated toward the obsessive. Lou Bloom, the unnervingly mannered stringer he played in 2014’s “Nightcrawler,” is a man of unhinged ambition. Georges Seurat, the post-impressionist painter Gyllenhaal portrayed in the 2017 Broadway revival of “Sunday in the Park With George,” is all about artistic perfection. In David Fincher’s 2007 fact-based procedural “Zodiac,” he played a cartoonist consumed by his pursuit of a serial killer. Gyllenhaal’s twitchy detective in 2013’s “Prisoners” is similarly driven by justice.

They’re all focused characters inhabited by a focused actor. But consider Gyllenhaal’s performances in the apocalyptic blockbuster “The Day After Tomorrow,” the $100 million rom-com “Love & Other Drugs” and the Marvel romp “Spider-Man: Far From Home,” and it’s clear he can also have a good time in a box office darling. Amid such widely seen projects — not to mention feverish public interest in his dating history — Gyllenhaal adopted a mantra of keeping his personal life personal.

“Being a famous person has, obviously, its great pros, but it also has its great cons,” says Andrew Burnap, the Tony winner who plays Cassio in “Othello.” “I think he, in his life, is very aware of that, and his commitment is to just being a great artist.”

That dedication to craft extends to playing the guitar, honing his photography and learning French. (“Jake constantly mocked me about the way I speak English,” Villeneuve says with a laugh, “and he took the risk of trying to learn French.”) A passionate cook, he tends to take charge of the Gyllenhaal-Sarsgaard household’s Thanksgiving feast. (“He’s just coming over with hot plates and things through the door,” Sarsgaard says. “It’s wonderful.”) Last fall, Gyllenhaal and Sarsgaard met with celebrity chef Hugh Fearnley-Whittingstall at his River Cottage estate in Devon, England; Gyllenhaal calls it “probably the highlight of my year.”

“I like the practice of cooking because it’s a constant discovery and experience,” Gyllenhaal says. “Maybe it’s the love of interpretation.”

Gyllenhaal’s interpretation of Iago, the Machiavellian underling previously played by the likes of Kenneth Branagh, Daniel Craig, Mark Rylance, Christopher Walken and Ian McKellen, earned raves when reviews for “Othello” dropped Sunday night. Although the overall production drew a mixed critical reception, its $2.8 million week during previews was the highest-grossing for a play in Broadway history, not accounting for inflation. (Last week, the George Clooney-starring “Good Night and Good Luck” bested that mark.)

Described by Washington Post theater critic Naveen Kumar as “the production’s indisputable anchor,” Gyllenhaal stalks the stage with the air of a men’s-rights provocateur in Leon’s 2028-set revival. Soon after Iago spits “Be a man!” to one pawn, Gyllenhaal lays bare the pain of a villain whose famously inscrutable motivations seem to veer, in this iteration, toward feeling underappreciated by Othello.

“Jake is special because he’s in pursuit of truth, and he does it on such a deep level,” says the director Leon. “It’s not someone twirling their mustache, and it’s not somebody playing an evil guy. This is somebody who has found the humanity in the most obvious of villains.”

Before Iago undoes Othello, the antagonist reveals his deceitful intentions during a celebrated soliloquy. It was during this speech — at the March 19 matinee, at least — that Gyllenhaal delivered the line “I hate the Moor” not with the cold calculation of many an Iago before but tortured conflict.

“It’s the age-old story,” explains Gyllenhaal, whose commitment to the character included a Venetian makeover of his dressing room. “It’s Steinbeck-ian. It’s siblings. It’s family. It’s being human. It’s the voice in our head that tells us ‘you can’t do that’ or ‘don’t do that’ or ‘you’re not capable of that.’ I think Iago is also hurt, and you can’t forget that.

“The argument he makes to himself is laying groundwork for something he needs to be true. Because I think he loves the Moor. When he says, ‘My lord, you know I love you,’ I don’t think that’s some manipulative thing in my choice. And I do love Denzel, so I can’t not play that.”

It’s a zig when the audience might expect a zag from an actor who showcased a sly, cunning side as the Spider-Man baddie Mysterio and that stealthily monstrous “Nightcrawler” character. Having read Martha Stout’s “The Sociopath Next Door” when preparing for “Nightcrawler,” Gyllenhaal decided against portraying Iago with comparable callousness. Still, finding his Iago’s empathetic tenor was a process that spilled well into previews.

“I was playing it pretty aggressively early on,” Gyllenhaal says. “The words themselves, if you enunciate them sometimes too much, if you don’t go back and give them speed and also space and grace, they become very evil.”

More than once, Gyllenhaal cuts himself off and cautions that he could ramble all day about his onstage process. For all of his on-screen successes, his fixated mind takes particular pleasure in finding ways to intrigue, unsettle or enchant a theater audience.

“Sometimes people come from a career in TV and film and it takes a minute for them to navigate a different medium,” says Annaleigh Ashford, Gyllenhaal’s “Sunday in the Park” co-star. “But it’s part of his marrow. He’s so at home onstage. He is so gifted at the give-and-take with the audience, the communion that you share with the living, breathing people that are sitting there watching you.”

While Gyllenhaal has booked his next two starring roles — a supernatural M. Night Shyamalan flick and a sequel to 2024’s “Road House” remake — he’s already kicking around ideas for a return to the theater. “The feeling I get before I go out every night is no different from the kid in high school on that stage,” he says. “The wings still look the same. The intensity is still the same. It may be Broadway, but the joy is the same.”

When I bring up the idea of an overarching trajectory, he chuckles, circles back to spontaneity and answers the question with one of his own: “Have you discovered that I clearly have no idea?”

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