1Heat gas or charcoal grill. In large bowl, mix beef, steak sauce, salt, 1/2 teaspoon of the pepper and 1/2 cup of the onions. Shape mixture into 4 patties, 1/2 inch thick. In small bowl, mix mayonnaise and remaining 1/2 teaspoon pepper, set aside.
2Place patties on grill over medium heat. Cover grill; cook 11 to 13 minutes, turning once, until meat thermometer inserted in center of patties reads 160°F. During last 2 minutes of cooking, place buns, cut sides down, on grill until toasted.
3Spread mayonnaise mixture on cut sides of buns. Top with remaining 3/4 cup onions, the lettuce and burgers. Cover with top halves of buns.