In a 10-inch ovenproof skillet heat oil over medium heat. Add red onion, shallots (if desired), and sugar. Cook for 12 to 15 minutes or until tender and golden, stirring often. Remove 1/4 cup cooked onion mixture and set aside. Stir green onion, mustard, and thyme into skillet.
In a bowl beat together eggs, water, salt, and pepper. Stir in 3/4 cup of the cheese. Pour into skillet. Bake in a 375 degree F for 15 minutes or until set. Top with remaining cheese and reserved onion mixture. Cut into wedges. Makes 6 main-dish servings.