Steam or boil the chicken. (Preferably steam.) Reserve the cooking liquid and defat. Use the reserved stock to cook the rice in. When cooled, remove the chicken meat from the bones.
Simmer the rice in 4 cups of the reserved broth until most of the liquid has been absorbed (about 12-15 minutes). Add the frozen broccoli and continue to cook just until the broccoli has thawed.
Combine the cooked rice and broccoli with the cream of mushroom soup, cream of celery soup, the chicken meat, milk, salt, pepper 1/3 of the onion rings (crumbled), and 1/3 of the Monterey Jack cheese.
Place mixture into one very large (or 2 smaller) buttered casserole dishes. Sprinkle the top with the remaining shredded cheese for about 30-40 minutes. Remove cover and continue to bake an additional 10 minutes or until browned and bubbly.
