Heat the oven to 375°F and coat a 9-by-13-inch baking dish with cooking spray.
Whisk the eggs, milk, dry mustard, salt and black pepper directly in the dish until no yolk streaks remain.
Fold in the frozen hash browns, sliced sausage and 1 cup cheddar, then spread the mixture evenly.
Cover with foil and bake for 35 minutes.
Uncover, top with the remaining cheddar and bake for 15 to 20 minutes, until the center reaches 160°F and no liquid egg moves when the dish is nudged.
Rest for 10 minutes before cutting into squares.