Melt butter in large skillet over medium heat. Add onions and cook until translucent, about five minutes. Stir in sugar and balsamic vinegar. Reduce heat to medium-low, cover pan, and cook for 30 minutes, stirring occasionally.
Preheat grill to medium. Lightly brush red pepper slices with oil and grill for two minutes per side, or until tender with some grill marks. Remove red pepper from grill. Brush mushrooms with oil and season with salt and pepper to taste. Add mushrooms to grill, stem sides down, and cook for five minutes. Flip mushrooms and cook for another four minutes.
Top four mushrooms with equal amounts of red peppers, cheese, and spinach. Cover each with a remaining mushroom, stem side down so that the stem sides are facing each other. Cook for one minute or until cheese has melted. If using buns, heat them on the grill for one minute, or until toasted.
Serve stuffed mushrooms topped with caramelized onions.