Mist a 9-by-13-inch baking dish with cooking spray. If baking immediately, preheat oven to 350ºF.
In a skillet over medium heat, cook sausage, stirring and breaking up chunks, until no longer pink, 5 to 6 minutes. Transfer to a bowl. Drain skillet of all but 1 Tbsp. fat.
Add onion to skillet and cook, stirring for 3 minutes. Add bell peppers and mushrooms; sauté for 5 minutes. Transfer to a bowl; let cool.
Whisk eggs, milk, salt and mustard. Stir in bread, Cheddar, sausage and vegetables. Pour into dish. Bake until middle is set and edges have browned, 50 to 60 minutes, or cover with plastic and refrigerate for up to 24 hours, then bake.