Heat a medium nonstick skillet coated with cooking spray over medium heat. Add onion; cover and cook 5 minutes, stirring occasionally. Add sugar, 1/4 teaspoon salt, and 1/4 teaspoon pepper; cook, uncovered, 5 minutes or until tender, stirring frequently. Remove onion from pan, and keep warm.
Heat pan coated with cooking spray over medium-high heat. Add mushrooms and 1/4 teaspoon salt; saute 5 minutes or until tender.
Combine 1/4 teaspoon salt, 1/4 teaspoon pepper, Worcestershire sauce, and beef. 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 5 minutes on each side or until done. Place bread on grill rack; grill 1 minute on each side or until toasted.
Arrange bread on a baking sheet. Top each bread slice with 1 patty, 1/4 cup onion, 1/4 cup mushrooms, and 2 tablespoons cheese; broil 2 minutes or until cheese melts.