Heat a large skillet over medium heat, and cook sausage for 5 minutes, until browned, breaking apart with a spatula. If there is fat, spoon out, and discard all but 2 teaspoons. Add onion and mushrooms to pan, season with salt and pepper, and cook for 8-10 minutes, stirring often, until golden brown. Remove from heat, and set aside.
Whisk together milk, Dijon mustard, dry mustard, nutmeg, eggs and a pinch of salt and pepper in a medium bowl.
Spray a 9×13 baking dish with olive oil or cooking spray, and arrange half of the bread slices along the bottom. Spoon half of the sausage mixture evenly over the bread, followed by 1/4 cup of cheese. Top with another layer of remaining bread, sausage and 1/4 cup cheese. Pour egg mixture evenly over the top. With the back of a spatula, press everything down to moisten with liquid. Cover with foil, and refrigerate overnight.
The next day, preheat oven to 350℉.
Top with the remaining 1/4 cup cheese, and bake uncovered for 40 minutes. Let stand 10 minutes, before slicing and serving.