Steam broccoli until tender (don't overcook) and rinse with cold water. Melt butter in a large saucepan and stir in the flour until smooth. Add the milk slowly, stirring constantly. Cook and stir until mixture thickens. Add the cheese and sour cream and continue cooking until cheese is melting, stirring constantly. Add ham, mushrooms, and broccoli. Place in casserole pan and heat in oven until hot and bubbly. Serve over hot cooked rice.