Corned Beef Casserole

8 ounces egg noodles, salted, cooked
    and rinsed in cold water
1/4 pound sharp Cheddar cheese, grated
1 cup milk
3/4 cup buttered bread crumbs
1 (12 ounce) tin corned beef,
    broken into small pieces
1 can cream of mushroom soup
1/2 cup onions, chopped and saut