1Melt butter in heavy medium skillet over medium heat. Add onions, bell peppers, and mushrooms. Cook until tender, stirring occasionally, about 7 minutes. Season to taste with pepper.
2Position rack in center of oven and preheat to 325 degrees F. Generously butter shallow 8-cup baking dish.
3Sprinkle shredded cheese evenly over bottom and sides of baking dish, pressing lightly.
4Process eggs, cream, milk, salt, cumin, thyme and hot pepper sauce 5 seconds in food processor (or whisk well by hand for about 1 minute).
5Set aside 1/3 cup of the vegetable mixture; add remainder to egg mixture and mix until combined (1 or 2 on/off turns if using food processor).
6Pour egg mixture into prepared dish. Bake until eggs are almost set in center, about 30 minutes. Sprinkle reserved vegetables over the top. Bake until eggs are just set, about 10 minutes longer. Serve hot.