Brown the sausage and onion in a large skillet over medium-high heat until the meat is no longer pink (about 7 minutes); drain.
Grease a deep 13 x 9-inch baking dish. Arrange bread cubes in the bottom of the dish. Add the sausage on top, followed by the cheese.
In a large bowl, whisk together eggs, milk, dry mustard, kosher salt and pepper.
Pour egg mixture over the casserole. Cover tightly with foil and refrigerate for 8 hours (or overnight).
In the morning, allow the dish to rest on the counter for 30-60 minutes before baking. Preheat the oven to 350 degrees F.
Remove the foil and bake the casserole (uncovered) for 60-80 minutes, or until the eggs are set and a knife inserted in the center comes out clean. Loosely cover the casserole with foil about halfway through, or whenever it begins to brown if you don't want the top to get too dark. The casserole will puff up very high in the oven, but will “deflate” significantly as it cools – this is normal.
Let stand 10 minutes, and then slice into squares and serve.