Sprinkle both sides of eggplant with salt and lay between double thick layers of paper towels. Press firmly and let stand for 15 minutes. Preheat oven to 450 degrees. Discard paper towels. Place eggplant on a large baking sheet and roast for 10 minutes, turning once.
Meanwhile, melt 1 tsp butter in a medium skillet. Add onion and cook until very soft, about 10 minutes. Remove from skillet and keep warm. Spread remaining butter on bread slices and place buttered side down in skillet. Place a piece of cheese and eggplant on each bread slice, then top with onions and roasted pepper. Cook for about 5 minutes or until cheese is melted and bread is golden brown. Carefully close sandwich.