Heat a large skillet over medium heat. Add oil to pan. Add sliced onion and 1/8 teaspoon salt; cook 18 minutes or until golden, stirring occasionally.
Combine 1/8 teaspoon salt, beef, and next 4 ingredients; toss gently. Divide mixture into 4 equal portions, shaping each into a 1/2-inch-thick patty.
Place patties on a grill rack coated with cooking spray; grill 4 minutes on each side or until desired degree of doneness. Top each patty with 1 tablespoon cheese during last 2 minutes of cooking.
Place rolls, cut sides down, on grill rack; grill for 1 minute. Place bottom halves of rolls on 4 plates; top each serving with 1 lettuce leaf, 1 tomato slice, 1 patty, 1/4 cup onion mixture, and 1 roll top.