Preheat the oven to 350°F (175°C).
Steam the broccoli florets until tender but still crisp, about 4-5 minutes. Drain and set aside.
In a large mixing bowl, combine the sour cream, condensed cream of mushroom soup, mayonnaise, shredded cheddar cheese, shredded mozzarella cheese, grated Parmesan cheese, minced garlic, chopped onion, salt, black pepper, and dried thyme. Mix well until fully combined.
Add the cooked chicken, steamed broccoli, and cooked white rice to the creamy mixture. Stir gently to combine all ingredients evenly.
Transfer the mixture into a greased 9×13 inch casserole dish and spread it out evenly.
Melt the unsalted butter in a small bowl and mix it with the crushed buttery crackers.
Sprinkle the cracker topping evenly over the casserole.
Bake uncovered in the preheated oven for 35 to 40 minutes, or until the casserole is bubbly and the topping is golden brown.
Remove from the oven and let it cool for 5 minutes before serving.