Alex Cooper Went From Raunchy Podcaster to Gen-Z's Barbara Walters

“I’m an open book,” Alex Cooper, the host of the wildly popular Spotify podcast Call Her Daddy, proclaims at the start of our conversation. It’s an afternoon in late May, and she’s lounging in a tasteful écru chair on the ground floor of her recording studio in West Hollywood, her two goldendoodles Bruce and Henry occasionally yapping at her feet. In ripped jeans, Adidas, and a white tank top with the slogan “your manifestations are on the way,” she looks like a college junior en route home from spring break, albeit one with flawlessly conditioned hair and a giant five-carat oval-cut diamond engagement ring on her finger. (Her finacé, YA film producer Matt Kaplan, proposed to her a few months before our interview, in a scavenger hunt leading up to a photo shoot, which she posted to her 2.4 million followers on Instagram.)Cooper has built a multimillion dollar brand around emanating candor; in the early years of Call Her Daddy, she chatted freely about everything from her specialized or...