Preheat oven to 400 degrees. In a skillet, melt butter over medium heat. Saute onion until lightly browned and tender, 5 minutes. Set aside. Grease 6 jumbo muffin tins with extra butter found in the skillet next to the onions.
Grate baked potatoes into a large bowl.
Stir in onions, remaining butter, egg, salt, and pepper.
Divide evenly between muffin tins and press evenly across the bottom and up the sides.
Bake 15 minutes or until edges are dark brown and crispy. Cool 2 minutes and remove from pans. Fill with scrambled eggs and serve warm.