The actress portrayed Daenerys Targaryen on the hit HBO series from 2011 to 2019

Emilia Clarke is looking back at Game of Thrones.

While chatting with PEOPLE about her new partnership with Pegasus Distillerie, the actress, 37, also recalls her time starring in the iconic HBO series, in which she portrayed Daenerys Targaryen, and how her feelings about the show have changed over the years.

Noting that it "means the world" that fans have such an attachment and appreciation for Game of Thrones, Clarke says she has been able to experience the same feelings now that the series has ended.

"The more distance I have from Game of Thrones, the more I can quantify it," she explains. "... When I started, you don't know what you're doing, you don't know what you're surrounded by and you don't know what you're taking part in."

"Now, as more and more time goes between it and me doing it, the more I'm like, that was incredibly special and that was incredibly rare," Clarke adds.

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Related: Prince William Meets Queen Daenerys! See Emilia Clarke Honored by the Royal at Windsor Castle

Game of Thrones — based on the series of fantasy novels by George R. R. Martin — ran for eight seasons from 2011 to 2019.

Alongside Clarke, others who starred heavily in the series included Lena Headey, Nikolaj Coster-Waldau, Kit Harington, Peter Dinklage, Sophie Turner, Maisie Williams and Isaac Hempstead Wright, to name a few.

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During the show's run, Clarke was nominated for four Emmys — in the outstanding supporting actress in a drama series category in 2013, 2015 and 2016, and in the outstanding lead actress in a drama series category in 2019.

Other award nominations that Clarke amassed during the series' run included nods at the SAG Awards, Saturn Awards, People's Choice Awards and the Critics' Choice Television Awards. She was also named one of TIME's 100 Most Influential People in the World in 2019.

Related: Game of Thrones: All the Spinoffs in the Works and What We Know So Far

Clarke recently founded her own production company, Magical Thinking Pictures, and said that the experience helped open her eyes to the behind-the-scenes processes that made Game of Thrones what it was.

"Oh my God, I really understand how the sausage is made now," the actress tells PEOPLE with a laugh.

Detailing that "sometimes" it is done "in a way that is quite frustrating,” Clarke continues. "It makes me see even more just how lightning in a bottle that was and how fortunate I was to have that as part of my experience as an actor."

"I was just incredibly, incredibly lucky to have had that experience, 100 percent," she adds.

House of the Dragon, a Game of Thrones prequel series, returns to HBO on Sunday, June 16, at 9 p.m. EST.

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