Walking into the offices of his publicist a few minutes earlier, Broadbent - polite, the barest glimmer of a smile - had greeted me with all the enthusiasm of someone who had been asked to fill out his tax return.
Softly spoken, the traces of his Lincolnshire upbringing still detectable, he has the air of a man with something on his mind, but he's not about to say what. The interview, he says, is "part of the job".
Broadbent is back on stage for the first time in two years, playing Hans Christian Andersen in a new play by Martin McDonagh, A Very, Very, Very Dark Matter.
In a sense, all McDonagh's work for stage and screen - In Bruges, Three Billboards Outside Ebbing, Missouri, The Pillowman - deals with very dark matters, but this, it seems, is darker still. The poster for the new play shows a rather scary looking eye, staring out through a tear in a black curtain. But what's it about?
I'd asked to see the script, I say, but was refused. Virtually all I know is the publicity line describing it as "dangerous, twisted and funny".
"I think that's all they want you to know, really."
He gives a slight smile: "It's very funny, very dark, very violent, and raises a load of different issues ..."
Can he tell me some of those "sorts of things"?
"Mmmmm." There is a long pause. "The nature of creativity. The massacres in the Congo in the 19th century - and that's probably as much of a teaser as I can give."
Another pause.
"It's a costume drama. I can let you know that."
Broadbent, 69, started out acting in theatre. His parents ran an amateur dramatic society and his first role was as a 4-year-old in a production of A Doll's House - "I started with Ibsen", as he likes to say.
His appearances on stage dwindled in recent years as film and television took over. If something exciting comes along, he says - which this play is - he's interested.
Across theatre, television and film - from sitcom to drama, from Only Fools and Horses to Mike Leigh dramas, to Woody Allen's Bullets Over Broadway - he has long been Britain's most versatile character actor.
"Fairly flexible, I think is the thing," he says, when I ask why directors keep coming back for more.
"My whole thing from the word go has been to do something I haven't been doing recently."
By way of illustration he points to the extraordinary run of performances that established him as an international star in the early 2000s - John Bayley in Iris (for which he won an Academy Award for Best Supporting Actor); the corrupt politician "Boss" Tweed in Martin Scorsese's Gangs of New York; W S Gilbert in Mike Leigh's Topsy-Turvy and the horribly grotesque showman in Moulin Rouge! (with Bridget Jones' dad thrown in for good measure).
I would imagine, I say, he's an actor directors can rely on.
"Well, I turn up ... "
His mother Dee was a sculptress who died "in the 90s"; father Roy died when Broadbent was 22.
Roy Broadbent was an artist and furniture maker who, during World War II, set up an educational commune in the Lincolnshire village of Holton cum Beckering, where pacifists and conscientious objectors could learn agricultural skills, and where Broadbent was brought up.
He was educated at a Quaker school, "a very horrid boy", he says, who was asked to leave after being caught drinking beer.
Broadbent strikes one as a reticent, in many ways closed-off, figure. Some journalists, he says, have called him rather dour.
"I'm getting better at it, but I'm not particularly happy talking about myself, which is probably why people write that my sentences tail off and I don't finish them."
He is not, he allows, much given to self-examination. "Never really interested me ... I quite like the mystery."
Broadbent did not marry until he was 38 because he was busy building a career. "I prioritised that - not consciously, perhaps, but that's what I was doing."
His wife, Anastasia Lewis, was a theatrical designer and is now an artist. They have two children by her first marriage, both in their 40s.
The couple live in London and have a home in Lincolnshire, where he pursues his other passion, carving lifelike wooden figures. About 1m high, rough-hewn, dressed and bewigged, Broadbent's "people" have an unsettling appearance, and some will be on stage in A Very, Very, Very Dark Matter.
He's looking forward to seeing them in period costume.
We have been talking for 45 minutes when his publicist pops her head around the door to tell him his car has arrived. The look on his face can only be described as relief.
