1. Heat oil in a large nonstick skillet over medium-high heat. Add onion, sugar, salt, and pepper to pan. Cook 6 minutes or until lightly browned, stirring occasionally. Add vinegar to the pan; cook 30 seconds, stirring constantly.
2. Combine cheese and next 5 ingredients (through beef) in a medium bowl; shape meat mixture into 4 (3-inch) patties. Heat a grill pan over medium-high heat. Coat pan with cooking spray. Add patties to pan. Cook 5 minutes on each side or until desired degree of doneness. Place 1 patty on bottom half of each bun; top each patty with 1/4 cup onion mixture and top half of each bun.