Preheat oven to 400°. Brush 2 ½-quart buttered casserole with melted butter. Add butter, milk and eggs to mashed potatoes; mix well. With a rubber spatula, create a soft fluffy texture. Place a spoonful of potatoes on a buttered sheet pan. Smooth with a spatula to create a flat surface. Place twelve ramekins into the mixture. Fill each ramekin with the potato mixture. Brush the top with the remaining melted butter. Bake for 30 minutes, or until golden brown. Place the ramekins in a baking dish and brush with egg and cheese mixture. Bake for 45 minutes, or until golden brown.