Bring a large saucepan of lightly salted water to a boil. Cook chicken in boiling water until no longer pink in the center and the juices run clear, about 30 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Remove chicken to a cutting board to cool. Shred cooled chicken into strands.
Preheat oven to 400 degrees F (200 degrees C).
Steam broccoli in microwave oven until tender-crisp, 4 to 6 minutes or according to package directions.
Stir mushroom soup and mayonnaise together in a bowl; add broccoli, onion, and mushrooms and stir to coat.
Spread cooked rice into the bottom of a casserole dish. Top rice with shredded chicken. Spread the mushroom soup mixture over the chicken. Sprinkle Cheddar cheese over everything.
Bake in preheated oven until slightly browned on top, 35 to 45 minutes.