Preheat oven to 350 degrees F. In a very large skillet heat oil over medium heat. Add onion; cook 5 minutes or until tender, stirring occasionally. Add chicken and mushrooms; cook and stir 5 minutes. Sprinkle with flour; cook and stir 2 minutes.
Stir in evaporated milk. Cook and stir until thickened and bubbly. Stir in broccoli; cook and stir 3 minutes more. Stir in the next six ingredients (through salt).
Transfer mixture to a 2-quart square or rectangular baking dish. Sprinkle with cheese. Bake 20 minutes or until bubbly and cheese is melted.