"What was my death like?" she asks -meaning her death as Miss Havisham in the BBC's new adaptation of Great Expectations, which she filmed earlier this year but hasn't seen yet.
Having just watched all three episodes, I can tell Anderson that her death, in a massive tower of flame, is pretty spectacular: shocking, moving, but most of all intriguing because it signals a new take on one of literature's most famous characters.
In Charles Dickens's original novel, the eccentric bride-that-never-was dies accidentally when her wedding dress catches fire by the hearth; but in this new version, she starts the fire herself. Miss Havisham appears to go willingly into the flames, to commit suicide. The woman who's been trapped in a living death has finally decided she wants to die for real.
Which leads to the start of my questions for Anderson. Later, I want to ask how an actor who used to be America's biggest TV star ended up in Britain, appearing in all these costume dramas, but first I want to get to the bottom of her take on Miss Havisham.
The most obvious innovation is one that Anderson can't help: the fact she's 43, which makes her the youngest Miss Havisham ever. The character – who befriends the hero of the novel, Pip, as he is raised to become a gentleman – is usually played as a woman in her seventies or eighties, but Anderson's version is a middle-aged woman made made frail by grief, not time.
"She comes across as even younger than I am," says Anderson, "and, certainly, the casting of me as opposed to somebody in their seventies was an intention of the producers and the writer Sarah Phelps not to write her as old as she's been written before. That was also how I heard her in my mind as I was reading the script." And there is another change Anderson wanted to make to the character – or at least the way she has sometimes been played in the past – and that was to play her much less as a bitter woman out for revenge and more as a sympathetic character, a woman we should feel sorry for despite everything she does to Pip. She still has the desire for revenge ("I want to hurt everybody," she spits at one point) but there's another side to her, and that's the woman who's been in pain so long it's become a way of life ("the agony is exquisite, is it not?" she says to Pip).
"My understanding of her from the book," says Anderson, "is that she is more directly nasty, and I really wanted to not just play that single note. I thought it might be more interesting that the damage she was causing was under a thicker veil. Until the moment she knows she's breaking Pip's heart and his dreams, there is a gentleness and a weight of heartbreak, so I hope there's a confusion of sympathy for her. I'm hoping that, on the one hand, your brain is going: 'Oh my God, what is she doing to these poor people?' and, on the other hand, it's feeling sorry for her."
The result is a portrayal of Miss Havisham that will probably divide the audience: some will see it as ghostly and grey and excitingly differently, others as mannered and possibly even over the top, especially the sing-song voice Anderson uses. Her aim, she says, was to try to catch the complexity of the character in the original book. There are so many different interpretations you can put on Miss Havisham's intentions – and this one is Gillian Anderson's.
So when did the actor first discover Dickens? "I didn't read him as a child," she says. "The first attempt I had at Dickens was probably when I was at college in my twenties. I think I probably struggled with him because I'm not sure I ever finished A Tale Of Two Cities. One has to have a certain amount of patience and intelligence even to get through his character descriptions. I discovered Dickens through acting."
Now, of course, Anderson has a strong, public relationship with Dickens, having appeared not only in Great Expectations, but also as Lady Dedlock in the BBC's take on Bleak House in 2005. It was a performance that rightly won Anderson a lot of praise, but it is just one of the recent historical roles she has taken on in Britain, including Wallis Simpson in Any Human Heart and Mrs Castaway in the recent dramatisation of Michel Faber's The Crimson Petal And The White.
Which means it's time for that other question: how come the actor most famous for The X Files has ended up with this rather unexpected career in British historical dramas? Is it because television over here offers better opportunities than it does in America?
"I've found it does," says Anderson. "There's nothing I've read in American television that has compelled me to do it in the last few years. I think I search out unpredictable characters, and they tend to be drawn out more slowly and in a more interesting way in the UK than how I've come across things in the States. Maybe I'm not reading the right American stuff, but my life is over here and I want to work over here."
Anderson also thinks her recent career choices might have something to do with the fact her childhood was in Britain. Until she was 13, she lived in London before her family moved to Michigan, which means she remembers the programmes she used to watch as a child: Blue Peter and The Magic Roundabout are two she picks out. It's left her with a strong affection for Britain and British life. "It's not so much an affection towards British television as an affection for British things. Britain feels like it's in my system."
There's another factor at play for Anderson, something else that has guided her towards British rather than American work, and in particular American movies: the fact she believes the characters she can play here are just better than the ones she could play in films. Modern film characters, she says, tend to be simpler, less complex and less interesting than characters like Miss Havisham: that confusing, opaque, baffling woman who, in this version at least, seeks out the friendly flames.
Anderson may not have seen that death scene yet, but she has seen the first episode of Great Expectations, which also stars Ray Winstone as the convict Magwitch. The famous opening chapter in which Pip encounters Magwitch for the first time on the marshlands is beautifully shot in a bright grey light; it's almost as if most of the colour has been drained out of it deliberately as a kind of tribute to the definitive David Lean film version of 1946.
"I haven't seen the David Lean film!" admits Anderson, and she obviously feels a bit guilty about that and tells me she plans to put it right by watching it over the Christmas holidays. "I'm embarrassed by the fact I haven't seen any of the other versions, but once I'd decided to do this project, I was conscious not to watch anything."
Even so, Anderson is still confident this adaptation is a good one because it captures her feeling that Great Expectations is the most modern of Dickens's novels. "The characters are so recognisable even in contemporary times," she says, "and the themes are still in play: class, greed, grief, guilt and heartbreak."
And tragedy. That's the theme that seems to have stuck in Anderson's brain: the tragedy of Pip and the fact he turned his back on the one person who loved him unconditionally. We may never know who this version of Miss Havisham really is, but we do know that for Anderson, that's what Great Expectations is really about.
Great Expectations, BBC One, December 27-29, 9pm.