Heat a large skillet over medium-high heat and stir in sausage. Cook and stir until sausage is crumbly, evenly browned, and no longer pink, about 10 minutes. Drain and discard any excess grease.
Beat eggs in a large bowl; stir in milk, salt, and mustard.
Fold bread cubes, Cheddar cheese, and sausage into the egg mixture.
Pour into a lightly greased 7x11-inch baking dish. Cover and refrigerate, 8 hours to overnight.
Preheat the oven to 350 degrees F (175 degrees C). Remove casserole from the refrigerator 30 minutes before baking.
Bake in the preheated oven until a knife inserted near the center comes out clean, about 40 minutes.