Preheat oven to 350 degrees F (175 degrees C). Grease a half-casserole baking dish or deep-dish pie plate.

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Heat a large skillet over medium-high heat. Cook and stir sausage in the hot skillet until browned and crumbly, 5 to 7 minutes; drain and discard grease.

Stir hash browns, cooked sausage, Cheddar cheese, 1/2 teaspoon salt, and black pepper together in a large bowl. Whisk milk, eggs, and remaining salt together in another bowl. Pour hash brown mixture into prepared baking dish; pour egg mixture evenly over hash brown mixture.

Bake in the preheated oven until bubbling and golden, about 45 minutes.