Preheat the oven to 350 degrees F. Melt the butter in a large skillet. Saute the onions and mushrooms in the butter.
Boil green beans in chicken broth for 10 minutes and drain. Add the green beans, mushroom soup, onion rings, and salt and pepper, to taste, to the onion mixture. Stir well. Pour into a greased 1 1/2-quart baking dish.
Bake for 20 minutes, then top the casserole with the cheddar cheese and bake for 10 minutes longer, or until the casserole is hot and cheese is melted.