'We're not the brand for people who eat nuts and bark," CKE Restaurants CEO Andy Puzder, tells me without apology. "Hell, I'll walk into one of our stores to get a barbecue chicken sandwich and walk out with a Western bacon cheeseburger. You smell them, and you just get hungry. We're the 'young, hungry guy' brand."

Even those who have never frequented one of Mr. Puzder's fast-food restaurants—known as Carl's Jr. west of the Rocky Mountains, and as Hardee's in the Eastern U.S.—will be familiar with his dude-centric...