The rapidly growing pack of burger chefs is sparking fierce competition to expand, protect innovations and promote their recipes as the world’s best. Most of the chefs make a big deal about the kind of meat served at their restaurants. Mr. Lagasse blends ground chuck, short rib and brisket; others promote their Angus, Kobe or grass-fed beef. Some beef experts say the main secret behind tasty celebrity-chef burgers is simple: They pile on the fat, whether from beef patties with 30% fat content or from patties basted in butter. That alone may make their burgers delicious at a time when supermarket ground beef may contain as little as 8% fat.