Cook sausages in a large skillet over medium-high heat, stirring until sausage crumbles and is no longer pink. Drain well.
Prepare hash browns according to package directions, using 1/2 teaspoon salt and pepper.
Stir together hash browns, sausage, and cheese. Pour into a lightly greased 13- x 9-inch baking dish.
Whisk together eggs, milk, and remaining 1 teaspoon salt. Pour evenly over potato mixture.
Bake at 350° for 35 to 40 minutes.