Step 1 Heat the oven to 350 degrees. Bake the sausage until it's browned and cooked through, about 30 minutes. Cut it into thin slices and set aside.
Step 2 Trim the mushroom stems and remove the gills from the caps by gently scraping them with a paring knife. Slice the caps and set them aside.
Step 3Heat 1 tablespoon of oil in a skillet over low heat. Add the onion slices and cook them until they're lightly browned, 25 minutes. Add salt to taste. Remove the onion from the skillet and set aside. Heat the remaining 2 tablespoons of oil in the skillet over low heat. Add the mushrooms and cook until they're dark brown, 5 to 7 minutes. Add the thyme and salt to taste. Set aside.
Step 4Mix together the eggs, milk, pepper and mustard in a bowl.
Step 5Spray a 13x9-inch pan with nonstick spray. Layer half the sourdough bread in the bottom of the pan, cutting it, if necessary, to make it fit, followed by a layer of sausage slices, onions, mushrooms, half of rye bread, diced, and half of the cheese. Pour half of the egg mixture over the top of the bread mixture. Repeat layering with the sourdough bread, sausage, onions, mushrooms and diced rye bread and ending with the cheese. Pour the remaining egg mixture on top. Press down to lightly soak the bread.
Step 6Cover the pan with parchment paper and foil and bake 30 to 40 minutes. Uncover the pan and bake until the strata is puffed and golden brown, 10 minutes more. Cut it into slices to serve.