Prepare grill to medium-high heat.
Combine balsamic vinegar, olive oil, and garlic in a large bowl. Add bell peppers and portobello mushrooms; toss gently to coat. Remove vegetables from vinegar mixture, and discard vinegar mixture.
Place bell peppers and mushrooms on a grill rack coated with cooking spray; grill 4 minutes on each side. Remove vegetables from grill; cool slightly. Cut bell peppers into thin strips. Combine bell peppers, basil, salt, and black pepper in a small bowl.
Cut rolls in half horizontally; spread cheese evenly over cut sides of rolls. Arrange 1 mushroom cap on bottom half of each roll; top each serving with about 1/3 cup bell pepper mixture and top half of roll.
Place sandwiches on grill rack coated with cooking spray. Place a cast-iron or other heavy skillet on top of sandwiches; press gently to flatten. Grill 3 minutes on each side or until bread is toasted (leave cast-iron skillet on sandwiches while they cook).