There’s no stopping Mildred Hayes, the mighty heroine of Three Billboards Outside Ebbing, Missouri. She’s out on a mission to find her daughter’s killer, swinging down Main Street like a pint-sized wrecking ball. She’s upending social niceties, trampling taboos and liable to flatten anyone in her path. As embodied by Frances McDormand, Mildred is the perfect emblem for the age of Harvey Weinstein and #MeToo – a hard-bitten, fiery angel of vengeance. Yet, tellingly, this virago is also the creation of a male writer-director. And just maybe he sees something of himself in her, too.

Like Mildred, Martin McDonagh can be a provocative, polarising figure. An upstart playwright turned film-maker, his work rides roughshod over Ireland, the US and the historic city of Bruges, exploding in a splash of bloody violence and plumes of pungent dialogue. One suspects that if he can’t be adored, he’d just as soon be loathed. “I’m coming from a punk-rock background,” he says by way of introduction. “The Clash and the Pogues. It’s all about trying to shake things up.”

Let’s start with the positives. Three Billboards Outside Ebbing, Missouri is his finest film to date, a rollicking carnival of US small-town life, managing to be pointed and profane but curiously compassionate, too. Enraged by the lack of leads in her daughter’s murder investigation, Mildred rents a trio of dilapidated hoardings as means of shaming the cops into action. Police chief Willoughby (Woody Harrelson) is embarrassed and the townsfolk are appalled. But Mildred is unrepentant. So what if Willoughby has terminal cancer? Who cares that the billboards have made her a social pariah? Mildred demands justice and feels that no price is too high.

Three Billboards premiered to rave reviews at the Venice film festival and went on to scoop the audience award at Toronto. This week, the film swept the board at the Golden Globes, winning four prizes including best film (drama) and a best actress (drama) gong for McDormand, whose performance is also being tipped for an Oscar. And so McDonagh rolls into London fully flushed with success; a softly spoken man with watchful eyes and a wolfish smile. He says he is delighted with the way the film has been received. He wrote the script with McDormand in mind and could never envisage anybody else in the role. “As a kid, I used to idolise actors like Marlon Brando and James Dean,” he says. “I used to try to model myself on those men. And little girls just don’t have those kind of characters to emulate – nobody that you’d want to walk and speak like. So this feels like a good film to be putting out at this time. It’s a good character to have walking out into the world.”

So far, so good. And yet, this being McDonagh, it hasn’t all been plain sailing. In recent weeks, critics have accused the director of a boneheaded approach to US race relations, highlighting his depiction of police officer Dixon (Sam Rockwell), a racist, trigger-happy mummy’s boy. In a more conventional (some might say responsible) film, Dixon would be painted the tale’s irredeemable villain. Audaciously, McDonagh suggests that he may not be all bad.

“It’s a big ask,” he admits ruefully. “I guess we’ll see just how big an ask.”

Actually, I tell him, concerns have already been raised.

McDonagh nods. “Yeah, I’ve seen that. I’ve read things. And I can totally see where the backlash is coming from. But I don’t think I redeem him, or forgive him, or try to make him a hero because the point is that there are no heroes or villains. The film isn’t about good and bad, left and right. It’s just trying to find the spark of humanity in people – all people. And that’s what it should be, or why start a film?”

Perhaps McDonagh thrives on controversy; it may be the water he swims in. He was raised in south London, the son of working-class Irish immigrants, and his writing reflects this mongrel pedigree; welding his family’s cultural roots with the extrovert drama of the US cinema he loves. His early rush of stage plays (The Cripple of Inishmaan, The Beauty Queen of Leenane) electrified the London theatre scene, but they also drew disapproval from Irish critics, who saw him as a fraud or a tourist; trading in stock “Oirish” stereotypes and treading a terrain he did not understand.

“Yeah, that did piss me off a little bit,” he says. “There was this idea that someone with my accent couldn’t call myself Irish, or that travelling back there every summer as a kid didn’t qualify you to talk about Ireland. But I think I’ve got past that now.” He grins. “And, actually, even at the time, it didn’t really stop me.”

There is a line in one of those early plays, The Pillowman, in which the lead character says: “The only duty of a storyteller is to tell a story.” McDonagh explains that he stole the quote from Orson Welles, but he reckons it sums up his ethos. “And there’s another line from the same play. Something like: ‘People say you should only write what you know. But you only write what you know because you are too fucking stupid to make anything up.’ And, yes, as it happens, I agree with that, too. Because writing is about lying, telling tales. You’re taking people to places they hadn’t thought about before.”

McDonagh’s writing is so flamboyant, so ornate, it must push his actors to their limits. I wonder if he ever feels the need to pare it back, tone it down. He laughs at the notion. “In all honesty, no. Maybe I should. But it’s not supposed to be a kitchen-sink drama. It’s a piece of cinema, or theatre, so the dialogue needs to be a little out there. We’re not recording the man on the street because that would be boring, at least to me. I’m not a kitchen-sink type of person.”

The thing is, he adds, he has always preferred cinema to the stage. He wrote seven plays in a mad rush over 10 months, but that was because he was bored and out of work, living alone at the old family home. He was as startled as anyone when he saw four of them running simultaneously on the London stage. McDonagh was the youngest playwright that had happened to since William Shakespeare’s day. So at the age of 27, he was an overnight success, embraced by a theatre establishment he had affected to despise. That must have taken a little getting used to.

McDonagh snorts. “Actually, it felt great. It was going from being completely on my own, not even having girlfriends, to being thrust into the world. Meeting people. Going to parties. But yeah, it was a big life change, obviously. It took some time to settle down.”

He recalls rolling up, drunk on vodka, to accept a prize at the Evening Standard theatre awards in 1996 and then swearing at Sean Connery when the actor ordered him to calm down. The tabloids had a field day; he was a pariah again. “That was the corrective. That set me straight. As soon as I got drunk and was a cunt to Sean Connery, everything suddenly fell back into place. Drunken Irish playwright. Briefly lauded and then: ‘Oh, no, not one of us.’ I think I’ve felt like an outsider ever since.”

I’m not sure I buy it. I think he’s safeguarding the brand. Over the past decade, McDonagh has won an Oscar for his short film Six Shooter and a Bafta for his acclaimed feature debut, In Bruges. He only stumbled slightly with 2012’s garbled, showboating Seven Psychopaths. “There were a lot of flaws with that film,” he says. “All of them mine.” Three Billboards looks well-placed for Oscar glory, plus he has recently completed a new play, A Very, Very, Very Dark Matter, which he describes as a “crazy, fucked-up life of Hans Christian Andersen”. So the future is bright; the man is a bona fide success.

McDonagh thinks back to that frenzied early period, writing scripts in a rush in his parents’ south London home. He wonders whether he will ever be so productive again – partly because he doesn’t really like working; partly because there are so many other demands on his time. “Sometimes, I wonder if I could just take a year out. Shut the door, get off the internet and write six scripts, one after the other.” For a moment, he looks very nearly wistful.

Since we’re taking a trip down memory lane, let’s briefly return to that drunken awards dinner. McDonagh hopped up on dutch courage, swearing at a startled Connery. I bet he feels embarrassed about that today.

McDonagh thinks it over for about a nanosecond. But the man has his limits; he’s not about to eat humble pie. He says: “No, not at all. It was totally brilliant.”

Three Billboards Outside Ebbing, Missouri is in cinemas now

• This article was amended on 18 January 2018. When McDonagh had four plays running simultaneously on the London stage, he was the youngest playwright that had happened to since William Shakespeare’s day, not the first playwright since Shakespeare as an earlier version said.