Whisk together eggs, milk and nutmeg in large bowl. Add bread cubes. Let bread mixture stand 30 minutes, stirring occasionally.
Heat oil in large nonstick skillet. Add onion, sugar, salt and pepper; cook over medium-low heat until onion is very tender, 15 minutes.
Meanwhile, heat oven to 350°F. Grease 13 x 9 x 2-inch baking pan.
Spoon half of bread mixture into prepared baking pan. Top with half of onion mixture. Repeat with remaining bread and onion mixtures. Sprinkle with cheese.
Bake in 350°F oven 45 minutes or until puffed and golden.
