Heat oil in a large nonstick skillet over medium heat. Add onion; cook 15 to 20 minutes or until golden, stirring frequently.
Remove brown gills from the undersides of mushrooms using a spoon; discard gills. Place mushrooms, cap side down, on baking sheet coated with cooking spray; broil 5 minutes.
Combine dressing and soy sauce. Brush mushrooms with mixture. Broil 5 minutes; turn and brush mushrooms with mixture. Broil 5 minutes or until mushrooms are tender. Top evenly with onions. Place 1 slice cheese over each, and sprinkle evenly with pepper. Broil 1 minute or until cheese melts.
Place mushrooms in rolls.