Preheat oven to 350 degrees.
Butter 9x13-inch pan. Cover bottom of pan with shredded hash browns.
Brown hamburger and onion. Place hamburger and onion on top of hash browns. Salt and pepper to taste.
Mix cream of mushroom soup and milk, pour over hash browns, meat and vegetables.
Place cheese slices on top and cover with aluminum foil.
Bake for 45 minutes. Uncover and sprinkle with onions, and return to oven with heat shut off for 15 minutes.