Whisk eggs, milk, parsley and salt; set aside. In a 9-in. cast-iron or other heavy skillet, cook sausage over medium heat until no longer pink; remove and drain. In same skillet, heat butter over medium heat. Add potatoes and onion; cook and stir until tender, 5-7 minutes. Return sausage to pan., Add egg mixture; cook and stir until almost set. Sprinkle with cheese. Cover and cook until cheese is melted, 1-2 minutes.