Make sauce by combining first 6 ingredients. Bring to a boil and cook for 1 to 2 minutes. Set aside.
Steam broccoli. While broccoli is cooking, heat oil. Sauté onion and minced garlic until garlic starts to brown. Add butter, then mushrooms and cook until almost done.
Add chicken; cook until heated through. Add broccoli. Pour desired amount of sauce over cooked ingredients. Cook for an additional 1 to 2 minutes, making sure sauce covers all ingredients.