Brush mushrooms with 4 tablespoons vinaigrette. Grill mushrooms and onion, covered, over medium heat 3-4 minutes on each side or until tender. Top mushrooms with red peppers, onion and cheese. Grill, covered, 2-3 minutes longer or until cheese is melted. Grill rolls, cut side down, 1-2 minutes or until toasted.
Spread roll bottoms with mayonnaise and drizzle with remaining vinaigrette. Top with mushrooms; replace roll tops.