Spray grill pan with cooking spray and place over high heat.
To prepare mushrooms for grilling: Remove the stems. Use a spoon to remove the gills by running it along the bottoms of the mushrooms. Wipe clean with paper towels.
In a small bowl, whisk together balsamic vinegar, olive oil, and salt and pepper to taste. Brush the mixture on both sides of each mushroom with a pastry brush.
When grill pan is very hot, place mushrooms in the pan, gill side up. Baste mushrooms with the vinegar and oil mixture. Cook for 8 minutes. Turn and cook second side for 6 minutes. Place a mozzarella slice on top of each mushroom cap and continue cooking for another 1 to 2 minutes to melt cheese.
To assemble: Place mushrooms, cheese side up, on a slice of bread. Evenly distribute roasted red pepper slices on the cheese and place a second slice of bread on top.
Serve with Quick Gazpacho.