That was back in 1996, since when things have panned out rather well for the Bathgate-born star of stage, TV and film. Whether such a weighty legacy puts any pressure on the cast of Jemima Levick's new Dundee production, however, remains to be seen.
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Whatever happens, as Levick resumes her role as the Rep's co-artistic director following 10 months on maternity leave, it is clear that Williams's first successful play, which premiered in 1944, remains close to Levick's heart. She first directed The Glass Menagerie in 2008, after all, in a production at the Royal Lyceum Theatre, Edinburgh.
Six years on - as she explains of her revisitation to the St Louis tenement where the dysfunctional Wingfield clan of Tom, his painfully shy sister Laura and their overbearingly optimistic mother Amanda retreat into their own worlds - there is unfinished business to attend to.
"Ever since I did The Glass Menagerie at the Lyceum I've wanted to go back to it again," Levick says. "There's so much in the play that I felt I never fully explored, and there was so much that I didn't have a chance to address, that it seemed like a good chance to go back to it now.
"One of the reasons I wanted to come back to the play is to look at Tom as its narrator. Given the play's autobiographical nature, the figure of Tom as a writer who is telling his story is really important. This is a man who ran away from everything, but who has come back because it still haunts him, and shows why he suffers so much."
Levick is also interested in the fact that The Glass Menagerie is not a naturalistic play, something she believes she didn't fully embrace at the Lyceum.
"This time round I want to give it more space, and embrace its dream-like quality, so that rather than it having a linear narrative, it comes from somebody's mind," she says.
"Tennessee Williams created an actors' version of the play after the first production, which had lots of explanations in it, but that didn't appeal to me. I much preferred the messy unstructuredness of the original script, but there are so many ways of doing it. I was looking at rehearsals, and I thought, you know, if I was any good as a choreographer, you could turn it into a really good dance piece. Maybe it'll become my thing to do The Glass Menagerie every five years or so in a different way."
While Robert Jack will play Tom in Levick's production, Tom's mother Amanda - a classic Williams matriarch full of self-delusional sass - will be played by one of the Rep's regular acting ensemble, Irene Macdougall. The latest incumbents of the theatre's graduate actor scheme, Millie Turner and Thomas Cotran, will be thrown in at the deep end to appear as Laura and Jim the gentleman caller, respectively.
"It's a fantastic opportunity," says Turner, who recently graduated from the Royal Conservatoire Scotland's acting course, having already appeared in the Dostoyevsky-based Notes From The Underground at the Citizens Theatre and #sleeptightbobbycairns at the Tron. "Laura is such a complex character. When the play was written, you might just have thought of her as shy, but now she seems to have more of a social anxiety disorder, and has created a world that excites her."
Cotran too is relishing the opportunity, especially as he graduated from the RCS's musical theatre course. "I'm really getting to delve deep inside Jim's character," he says. "He has to bring a new energy and be this breath of fresh air and positivity, which is in complete contrast to everyone else."
As Levick recognises, "this is a really big challenge for Millie and Tom, but it's one I know they're already rising to, which is why they're here. One of the great things about having the acting ensemble in Dundee is that there is a certain amount of pastoral care and support for each other, and because Millie and Tom are with the ensemble for a year, this is just the beginning of their journey, and we can build on whatever happens now".
At the end of October, Dundee Rep will continue the Tennessee Williams theme with Talk To Me Like The Rain And Let Me Listen, a compendium of four short Williams plays which will tour community centres in Dundee.
Directed by Macdougall and featuring Turner and Cotran in the cast, this quartet of impeccably realised and rarely seen miniatures features This Property Is Condemned, Mr Paradise, Auto-da-Fe and the beautiful title play, a production of which was seen alongside This Property Is Condemned in a similarly minded triple bill at the Tron Theatre in Glasgow as part of Glasgay! 2008.
Next up for Levick in Dundee will be the Rep's Christmas show, a new take on Roald Dahl's James And The Giant Peach. This should provide some welcome light relief after The Glass Menagerie, especially as once her production is up and running she will be visiting Oslo to take part in a festival of work by Norwegian playwright, Henrik Ibsen.
"From one jolly play to another," she jokes, before pointing out what continues to fascinate her about The Glass Menagerie.
"I think it's the people," she says, "the characters and their relationships, which are so truthful and compelling. If you break it down, it's a very simple look at the relationships between a mother and son, a brother and sister, a mother and daughter, and the people around them. That's beautifully realised, because these people are so hopeful, but are so disappointed as well, and I think there are parallels there with all our lives."
The Glass Menagerie, Dundee Rep, September 3-20. Talk To Me Like The Rain And Let Me Listen, October 20-November 1.