It took Martin McDonagh roughly 20 years to get a comedy about capital punishment to Broadway — but the delay was not for the reasons you might expect.

Mr. McDonagh, whose capacity for such a feat had been suggested by spectacularly entertaining and profoundly disconcerting plays like “The Beauty Queen of Leenane,” “The Cripple of Inishmaan,” and “The Pillowman,” was already working on his current Broadway outing, “Hangmen,” back in 2001. 

“I wanted to explore something about miscarriages of justice,” the playwright and screenwriter says on a call from London, where he was born to Irish parents and is still based. “I thought about that period in England before the abolition of hanging. It was a dark, grim time, just before the swinging ’60s. It seemed like ammunition to tell a creepy tale, but I wasn’t sure what genre to put it in. That might have been why I put it aside for a while; I thought it had potential as a thriller, but I wasn’t thinking about trying to write a thriller then.”

“Hangmen” finally made its world premiere to wide acclaim at London’s Royal Court Theatre in 2015, and then transferred to the West End and off-Broadway, where critics and audiences were similarly enchanted by its blend of gallows humor and hair-raising suspense. The Broadway opening was postponed because of the Covid shutdown, though it’s been just as well-received: “Hangmen” is up for five Tony Awards this Sunday, including best play, marking Mr. McDonagh’s fifth nomination.

Set in Northern England in the mid-1960s, “Hangmen” follows a veteran executioner, Harry, who is facing the extinction of his profession as questions re-emerge about the guilt of one of his previous victims. The arrival of a mysterious stranger from London unsettles matters further, particularly when the visitor takes an interest in Harry’s teenage daughter.

Mr. McDonagh was inspired in part by the work of Harold Pinter, an enduring influence. “I definitely wanted to capture the sort of menace you find in his early plays, which I think can be quite funny as well,” he says. 

Staging “Hangmen” in America, where the death penalty is still on the books in many states, has been “strange,” Mr. McDonagh admits. “It’s a historical piece in the U.K., but not here. That’s why I think it’s good to explore it here, to see if there’s any difference in people’s opinions. Of course, we’re in New York, which is pretty liberal; it would be interesting to see it in states that do have capital punishment.”

Granted, avoiding pedantry has always been a priority for Mr. McDonagh. “I do try to probe moral questions, but I think you can do that in the form of an entertaining story, or a funny one,” he says. “In fact, you can get away with a lot more if you do in that way, because you can express your opinions without being heavy-handed — or even seeming like you’re expressing opinions.” 

While Mr. McDonagh’s own views “fall to the left of the political spectrum,” increasing debates over language and its context have made him concerned that “we may be at a place where there’s confusion over what the character is saying and what the playwright is trying to say. Just in the past five years, I’ve had theaters wanting to do my older plays ask me to change lines. A couple of lines were considered racist, for instance — but they’re meant to be, because they’re spoken by racist characters.” 

Although Mr. McDonagh has declined such requests, he discovered that some dialogue had been altered without his permission upon attending an Australian production of “The Lieutenant of Inishmore,” an account of a maniacal Irish freedom fighter who becomes even more unhinged after his cat is killed. Such alteration “is actually illegal,” he notes. “That’s going way too far.”

“Hangmen” boasts its share of buffoons, and the relatively clever albeit sinister Londoner — whom the playwright describes as “the antithesis of me” — mocks paraplegics in one darkly hilarious exchange, and notes of Groucho Marx, “He was a Jew, but you couldn’t really tell.” Mr. McDonagh has “heard reactions in the audience, but I don’t think there have been any complaints or walkouts. I guess people understand that it’s not me expressing these thoughts.”

With the production closing June 18, Mr. McDonagh is putting the final touches on “The Banshees of Inisherin,” his first feature-length film since 2017’s Oscar-nominated “Three Billboards Outside Ebbing, Missouri.” Shot on the west coast of Ireland, the new movie reunites its screenwriter and director with actors Colin Farrell and Brendan Gleeson, who last appeared together in Mr. McDonagh’s 2008 film “In Bruges.”

Early reports described “Banshees” as the realization of an unfinished final chapter in Mr. McDonagh’s Aran Islands Trilogy, which also includes “Inishmore” and “Inishmaan.” “It almost feels like a third part,” he allows, “but in a very vague way. The story is set on an island, and the dialogue style isn’t dissimilar. But maybe I shouldn’t say more, except that I’m very happy with it. I think it’s very funny—and very sad.”