Pascal Le Segretain/Getty
While attending the Three Thousand Years of Longing afterparty at the 2022 Cannes Film Festival, which took place at La Mome Plage in France, Miller, 77, opened up to PEOPLE about his first encounter with the actress, who is known for fully transforming herself with each acting role she takes on.
Noting that he was familiar with Swinton's work, and that he had previously met the star years ago while at a dinner for the Cannes Film Festival's 70th anniversary, Miller told PEOPLE, "Because she's so different in any role from the movies, I had no idea who she was in person."
"But," the famed director — who has worked on 1979's Mad Max and 2015's Mad Max: Fury Road — continued, "she was incredibly impressive to meet."
Added Miller of Swinton, 61: "And the moment I met her that day, I thought, 'Oh God, it'd be great if she could play [this role we have for her].' So we went to her."
Three Thousand Years of Longing — an adaptation of the short story "The Djinn in the Nightingale's Eye" by A.S. Byatt — made its debut at the Cannes Film Festival on Friday.
In the film, academic Dr. Alithea Binnie, played by Swinton, lets a genie out of the bottle when she brings home a trinket from a bazaar in Istanbul while at a conference.
Never miss a story — sign up for PEOPLE's free daily newsletter to stay up-to-date on the best of what PEOPLE has to offer, from juicy celebrity news to compelling human interest stories.
Swinton's character soon finds herself with the ancient Djinn (Idris Elba) and three wishes on her hands. However, the sensible scholar of mythology is all too wise to the dangers of the possibilities before her, leading to some unexpected outcomes.
RELATED VIDEO: Idris Elba Takes on Hot Ones and Reveals the Moment He Felt Famous
Continuing to chat with PEOPLE, Miller also spoke about casting Elba, 49, and how the actor was approached about the role after Swinton was officially a part of the project.
"We met Idris for the first time at the BAFTAs, during Fury Road. And once again, that sort of quality really struck me," he said. "So once Tilda was cast, we went straight to Idris."
Casting the duo together, Miller noted, was a perfect match. "They proved to be wonderful," he shared.
"I mean, the way they work together, they're what I call filmmaking actors. They're there to get the film made and bring the best of themselves to it," the director added. "And they were just fantastic in it."
Three Thousand Years of Longing hits theaters Aug. 31.