Preheat oven to 350 degrees. Butter/grease a half-casserole baking dish or deep-dish pie plate or cast iron skillet.
Heat a large skillet over medium high heat. Cook and stir sausage and diced onion (if using) in the hot skillet until browned and crumbly, about minutes. Drain and discard grease.
Stir hash browns, cooked sausage, 1-1/2 cups 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 45-55 minutes. Top with remaining cheese as soon as it comes out of the oven. Sprinkle with green onions, if desired.