Crumble sausage into 10-inch nonstick skillet. Cook, stirring occasionally, over medium heat until lightly browned. Remove sausage with slotted spoon to medium bowl; discard all but 1 T. drippings.
Stir onion into skillet. Cover and cook over low heat until onion is tender, about 5 minutes. Add potatoes and stir to coat with drippings. Cover and cook over medium-low heat 6-10 minutes. Turn potatoes with spatula. cover and cook until edges brown and potatoes are tender, 6-8 minutes longer. Sprinkle with pepper and arrange potatoes evenly in skillet. Sprinkle with layer of sausage, then layer of cheese.
Beat eggs with milk and carefully pour over mixture in skillet. Cook covered over medium heat until eggs are set, 12-15 minutes.