In a pot boil chopped broccoli florets until tender. Remove and drain water. In the same (emptied) pot add the chopped onion, mushrooms, and butter. Saute until the onions are translucent. Add broth, cream of mushroom soup, salt, pepper, and broccoli. Bring to a boil. Add cheese and stir so that the cheese does not clump. Soup is ready to serve.