Preheat oven to 350 degrees F (175 degrees C).
Grease a half-casserole baking dish or deep-dish pie plate.
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, 1 c 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.
Sprinkle remaining cheese over the top.
Bake in the preheated oven until bubbling and golden, about 45 minutes.