Assembly. In a large skillet saute the onion, scallion, and mushrooms in 2 Tbsp oil over moderately low heat, until the onion is softened and all liquid the mushrooms give off is evaporated.
In another large skillet saute the bell peppers in remaining oil and salt and pepper to taste, until peppers are tender.
Combine vegetables and season.
Arrange half the bread cubes in a buttered large shallow (4 1/2-quart) baking dish, spread half the vegetable mixture over them, and sprinkle half the Cheddar and half the Parmesan over the vegetables.
Arrange the remaining bread cubes over the cheeses, top them with the remaining vegetables, and sprinkle the remaining cheeses over the top.
In a bowl whisk together the eggs, the milk, the mustard, the Tabasco, and salt and pepper to taste, and pour the egg mixture evenly over the strata.
Cover and chill the strata, overnight. Let the strata stand at room temperature for 15-30 minutes and bake it in the middle of a preheated 350°F. oven for at least 1 hour, or until it is puffed and golden and cooked through.