Heat broiler to high. Set one rack 6 inches from heat source. Place second rack as far from heat source as possible.
Place peppers on a rimmed baking sheet. Toss with 1 tbsp of the olive oil and 1/8 tsp each of the salt and pepper. Place on lower rack of oven and cook 15 minutes, turning once halfway through, until peppers are tender and slightly browned.
Meanwhile, clean mushrooms and remove gills. Brush with remaining 1 tbsp oil and place on a separate baking sheet, cap side up. Place on top rack and broil 4 minutes. Turn and season with remaining 1/8 tsp each salt and pepper. Broil an additional 3 minutes. Fill with tomatoes and top each mushroom with 3 tbsp cheese. Broil 3 to 4 minutes.
Sandwich mushrooms between bread slices and serve peppers on the side.