Preheat oven to 350°F (175°C).
Steam the broccoli florets until just tender, about 4-5 minutes. Drain and set aside.
In a large mixing bowl, combine the shredded cooked chicken, steamed broccoli, cream of mushroom soup, sour cream, mayonnaise, chopped onion, minced garlic, salt, black pepper, dried thyme, and cooked white rice. Mix well until all ingredients are evenly incorporated.
Fold in the shredded cheddar cheese.
Transfer the mixture into a greased 9×13 inch baking dish and spread evenly.
Sprinkle the shredded mozzarella cheese evenly over the top of the casserole.
Sprinkle the crushed buttery crackers evenly over the mozzarella cheese layer.
Cover the casserole tightly with aluminum foil.
For freezer meal preparation: Place the covered casserole in the freezer. When ready to bake, thaw overnight in the refrigerator.
Bake the casserole uncovered at 350°F (175°C) for 35-40 minutes or until bubbly and the top is golden brown.
Remove from oven and let stand for 5 minutes before serving.