Preheat oven to 350°F (175°C).
Lightly grease a 9×13 inch casserole dish.
In a large mixing bowl, combine the shredded cooked chicken, fresh broccoli florets, cooked white rice, chopped onion, and minced garlic.
In a separate bowl, mix together the condensed cream of mushroom soup, sour cream, mayonnaise, salt, black pepper, and dried thyme until smooth.
Pour the soup mixture over the chicken and broccoli mixture and stir until everything is evenly coated.
Add the shredded sharp cheddar cheese and shredded mozzarella cheese to the mixture and fold gently to combine.
Transfer the mixture to the prepared casserole dish and spread evenly.
In a small bowl, combine the melted butter, grated Parmesan cheese, and crushed buttery crackers.
Sprinkle the cracker topping evenly over the casserole.
Cover the casserole with aluminum foil and bake in the preheated oven for 30 minutes.
Remove the foil and bake for an additional 15 minutes, or until the topping is golden brown and the casserole is bubbly.
Allow the casserole to cool for 10 minutes before serving.