Place the sausage in a large skillet and heat. When it has begun to brown, add onions. Continue cooking until the sausage is cooked through and the onions are soft. Drain fat, if necessary. When done, place in a 10x14 baking dish (see notes) and allow to cool slightly.
Meanwhile, combine the eggs and milk in a large mixing bowl. Whisk together. Add the flour and seasonings and whisk well; it will be lumpy. Stir in the shredded cheese. Pour over the sausage in the baking dish.
Heat the oil in a skillet. Cook the hash browns in batches, seasoning with salt and pepper. Let cool slightly, then layer over the egg mixture in the baking dish. Press down lightly into the egg mixture.
Bake at 375°F for about 35-45 minutes, or until the egg layer is set in the middle.