Pre-cook chicken (great way to use up left over chicken or I usually boil the breasts and reserve the broth for future recipes); cut into bite sized pieces. Layer chicken in a greased 9 x 13 casserole pan. Place broccoli on top of chicken. Mix thoroughly the mushroom soup, mayonaise, cheese and onion. Pour over top of broccoli and chicken. Mix melted butter with cracker crumbs. Sprinkle evenly on top of casserole. Cover and bake at 350 for 45 minutes or until evenly warmed.