1. Heat the oil in a large skillet over medium-high heat. Add the onion, peppers, and shiitakes, along with ½ teaspoon of the salt. Cook, stirring from time to time, for 5-7 minutes, until the vegetables have softened, browned a bit, and decreased in volume. Remove from heat.
3. Crack the eggs into a large mixing bowl and beat well with a fork. Stir in milk, dry mustard, the rest of the salt, and a few grinds of black pepper. Add bread cubes and stir to moisten. Then stir in all the vegetables, the parmesan or pecorino, and three-quarters of the gruyere and cheddar. Pour the whole mixture into a 9 x 12 baking dish and sprinkle with the remaining cheese. Cover and refrigerate overnight or for at least a few hours before proceeding. Remove from fridge about an hour before baking if possible.
4. An hour before serving time, preheat oven to 375 degrees F with a rack in the center. Bake uncovered for about 40 minutes, until puffed and golden. Serve hot.