A little over a year ago, Patrick Schwarzenegger sat in his agents’ office to put together his career goals for 2024. He was coming off of a strong run of prestige television roles — HBO’s The Staircase, Seth Rogen’s The Boys spinoff Gen V — but there was one white whale that he couldn’t stop thinking about.

“At the very top of my list was wanting to work on White Lotus season three,” he tells The Hollywood Reporter. “It’s an actors’ dream, you do one single season, on a really great network, and you get to go and live on an island for six months. So I told them that, even though I knew it was out of my agents’ hands because we had no control over what kind of characters Mike White was going to write or what I would be good for.”

Of course, as White Lotus fans know,...