Cut bell pepper in half lengthwise; discard seeds and membranes. Place pepper halves, skin sides up, on a 15 x 10-inch jelly-roll pan coated with cooking spray; flatten with hand. Place mushrooms (gill side down) and onion slices on pan. Combine 1 1/2 tablespoons balsamic vinegar and olive oil, stirring with a whisk. Brush vinegar mixture evenly over mushrooms, onion slices, and pepper halves.
Broil 8 to 9 minutes or until mushroom caps are tender. Remove mushrooms; broil onion and bell pepper 3 additional minutes or until crisp-tender.
Place mushroom caps on bottom halves of buns; carefully place onion and pepper halves over mushroom caps. Place cheese over onion. Broil 1 minute or until cheese begins to melt. Top with basil leaves, and sprinkle each with 1/2 teaspoon vinegar. Cover with top halves of buns.