In large bowl, whisk together eggs, milk, pepper and nutmeg; stir in noodles to coat. Stir in onion mixture and half of the cheese. Place in buttered 13- x 9-inch (33 x 23 cm) baking dish. Sprinkle with remaining cheese. Bake, uncovered, in 375°F (190°C) oven until lightly browned, about 30 minutes.