Preheat the oven to 350°F (175°C).
Lightly grease a 9×13 inch casserole dish.
Steam the broccoli florets until tender but still crisp, about 4-5 minutes. Drain and set aside.
In a large mixing bowl, combine the cream of mushroom soup, sour cream, milk, shredded cheddar cheese, chopped onion, minced garlic, salt, black pepper, and dried thyme. Mix well until smooth.
Add the cooked rice, shredded chicken, and steamed broccoli to the bowl. Stir gently until all ingredients are evenly combined.
Transfer the mixture into the prepared casserole dish and spread evenly.
Sprinkle the crushed buttery crackers evenly over the top of the casserole.
Bake in the preheated oven for 40-45 minutes, or until the casserole is bubbly and the topping is golden brown.
Remove from the oven and let it rest for 5 minutes before serving.