In a large skillet, melt 1 tablespoon butter over medium heat; stir in the potatoes, sausage, green and red peppers, salt and cayenne. Cover and cook until potatoes and vegetables are tender, 12-14 minutes, stirring occasionally. Stir in cheese.
In a large skillet, fry eggs in remaining butter as desired. Sprinkle with pepper. Serve eggs with hash and, if desired, pepper sauce.