Benedict Cumberbatch doesn’t like to overshare, but sometimes he just can’t help himself.

Take the two hours we spend on a recent London afternoon discussing his new film, “The Thing With Feathers,” a dark drama about a father struggling to hold his family together after his wife dies suddenly. While the film paints an unvarnished portrait of a man in crisis, Cumberbatch feels awkward talking about how he channeled that kind of soul-eviscerating grief on-screen. “The Brit in me is a bit embarrassed about diving too deeply into what I do,” he admits.

There’s a polite reserve to Cumberbatch, as well as a discomfort with the trappings of stardom. Still, there are moments when his guard drops and suddenly he’s telling all. Cumberbatch is best known for playing Doctor Strange, the brilliant but arrogant Sorcerer Supreme in six Marvel movies. It’s the role that gave...