"Within an hour of reading it, he rang Matt and said, 'I'm in'," says producer Nick Batzias (who has a small role in the film as "Greek @ auction"). "He loved the script. There just aren't a lot of roles like this around for an actor, and Anthony has earned himself the right to pick and choose what he wants to do."

In AMOS, as the filmmakers shorthand it, LaPaglia plays real estate agent Frank Mollard, a man whose joie de vivre has taken something of a battering.

His wife (Justine Clarke) has gone from community-theatre nobody to hospital-soapie superstar, and Frank feels he has been left behind. His teenage son (Indiana Crowther) doesn't want to know him. His boss, estate agent Phillip Lang (a brilliantly funny John Clarke), is keeping him afloat, but even his saintly patience is being tested by Frank's inability to use the phrase "dream property" without dripping sarcasm.

Most pressingly, Frank has unresolved issues stemming from the death of his mother. Enter the elderly Sarah (Julia Blake) via a random wrong number and you have the makings of an unexpectedly warm, sad, funny comedic drama with a very loud ring of truth in spite of the implausibility of a premise that has Frank effectively adopting Sarah as his mother in order to fix the relationship he never had with his real one.

As Clarke's Phillip Lang puts it in one priceless moment, picking up a phone: "It's for you, Frank. It's Freud."

I sat down with LaPaglia on the set of A Month of Sundays on a glorious day in February 2015. Adelaide was bathed in gentle sunshine and the playing fields of Prince Alfred College, where the crew was filming, radiated good health and massive wealth. LaPaglia wore the standard-issue estate agent uniform of slightly crumpled, slightly shapeless dark-grey suit, though with the fetching addition of an adhesive name tag stuck to his lapel, with "Adolf" written on it in black Texta (further evidence of Frank's cheery state of mind).

For the best part of an hour we chat; he seems relaxed, at ease, if a little stunned to find himself back where it all began (he has of course made movies in Australia before – Lantana, Lucky Break and Balibo among them – but he has never before worked professionally in the city in which he was raised).

The night before, he'd had dinner with about 20 cousins he had never met before. "They were all born after I left," he says. "It was like giving a press conference. It was quite overwhelming, actually, there were so many of them, but it was lovely to meet them."

What wasn't known at the time was that LaPaglia and his wife of 17 years, actress Gia Carides, were in the throes of a break-up that would become public a few months later. Having made five films back to back, he tells me he intends to take some time off. "My daughter's starting to resent me. I need to go home." He also mentions his father had died the year before.

All of which might go some way to explaining just why he connected so strongly with Frank.

Writer-director Saville won't be drawn on his star's personal circumstances, but he says: "It was kind of terrifying how much both Anthony and I related to Frank".

Mostly, he says, what they recognised was "his bad behaviour. 'Yeah, I've done that, I've said that.' It was kind of incriminatingly easy for us. The third act, where people start behaving with dignity, was harder for Anthony and I. We had to imagine better versions of ourselves."

In a sense, that's what LaPaglia has been doing for decades, ever since he hung up the gloves on his fledgling career as a goalkeeper and decided to chance his arm in the States.

He was just 21 when he headed overseas, with a handful of acting classes in Adelaide under his belt and more ambition than strategy in his duffel bag.

"I would love to say that it was ballsy, but it was more sheer ignorance," he says of the decision to move to New York. "When I first decided I was going to do it I was too stupid to think it was impossible. I think you need to be a little bit dumb to be an actor."

He chose the US because it seemed a better place for someone of his heritage – Italian father, Dutch mother, working-class upbringing – to catch a break.

"At that time in the Australian industry, if your name ended in a vowel there weren't that many opportunities. You could be Con the fruiterer or Tony the petrol station attendant. And I just thought, 'If I'm going to do it, why not go big'."

For the first five years in New York he tended bar, took classes – four or five a week, as many as he could afford – and occasionally landed a role like Mechanic #2 (Amazing Stories, 1985); Punk #1 (The Twilight Zone 1986); or Agent #1 (The Equalizer, 1988).

His first big role was the lead in Frank Nittie: The Enforcer, a 1988 telemovie about an associate of Al Capone. By then, LaPaglia had learned to lie about being Australian, because the alternative was to have "a 20-minute discussion about shrimps on the barbie" followed by rejection. "Thank you Paul Hogan."

With the Nittie role, he says, "I'd done all the research, I knew he was from Riverside in Chicago. So when the producer said, 'Where are you from?', I said, 'Chicago'. He said, 'Where?', I said 'Riverside'. He went, 'That's amazing!' 'I was born to play this part.'

"It wasn't until about halfway through filming that he discovered the truth – and he was furious."

By then, though, LaPaglia was off, and in the decades since he's been consistently busy. But, he insists, "I've been hot and not 15 times – and what I've found is it's your ability to survive the downward cycle that separates you from the rest.

"I never thought 'Will I ever work again'. It was just, 'I think I'm on the downside; what can I do to get back on the upside?' Usually I'd get back to theatre. That would help."

In 2002, he landed the role that made him a household name in the US – as FBI missing persons detective Jack Malone in Without a Trace, which ran for 160 episodes over seven seasons.

"It was a long haul, doing the same thing eight or nine months a year," he says. "After it was over, there were two things: first, financially I could afford to sit back more; and second, I was tired – I was really tired – and I think I'd lost a little bit of the joy of acting by pumping out the same stuff every year."

At what point does it simply become a grind?

"About year five."

So what keeps him doing it?

"Your contract," he says, with his gruff smoker's laugh. "They will sue you to high heaven if you try to get out."

When it was done, he says, "I was quite relieved … I took some time off, a year and a half, maybe two years, spent some time with my daughter. And then I started to feel the enthusiasm again."

He certainly felt that buzz when he read A Month of Sundays but, he says, "A lot of things can happen between wrap day and post-production so I've learnt to lower my expectations."

Judging from the fact that he's touring the country for Q&A sessions of the movie we can assume he's pretty happy with how it turned out, though Saville says LaPaglia apologised to him after first watching it.

"He said, 'I didn't get it right. I went too far under the wire'," – that is, he felt he had played Frank in too restrained a manner. "But he did exactly what I asked.

"I think his performance is really brave," Saville continues. "There's no vanity. He didn't go for the easy laughs. He made the character difficult to like, but you do eventually. And when you do – hopefully – it has more meaning, more resonance, precisely because he didn't make it easy."

A Month of Sundays opens in cinemas on April 28.

Karl Quinn is on Facebook and on twitter @karlkwin