“The Girlfriend Experience” has all the hallmarks of a highbrow TV drama. The new series on pay-cable channel Starz features a flawed main character—a law-firm intern who moonlights as an expensive escort. It was produced by an Oscar winner, Steven Soderbergh, and “suggested by” his feature film of the same name. Two independent filmmakers, Lodge Kerrigan and Amy Seimetz, wrote and directed all 13 episodes. The first four episodes premiered at the Sundance Film Festival.

But the steely, sex-heavy drama also shares something...