Heat oven to 375. In a 10-12" cast-iron skillet, heat oil over medium heat. Add red onions, shallots and sugar. Cook until tender, about 10 min. Stir often. Remove 1/4 cup of the mix and set aside. Stir green onion, mustard and thyme into skillet.
In lg bowl, beat together egg, water, salt and white pepper until frothy. Stir in 3/4 cup cheese. Pour mix into skillet.
Bake 15 min, or until set. Top w/ remaining cheese and reserved onion mix. Cut into 6 wedges.