Preheat the oven to 350°F (175°C).
In a large skillet over medium heat, melt the butter. Add the diced onion, celery, carrots, and sliced mushrooms. Sauté for 7-8 minutes until vegetables are tender.
Add the minced garlic and cook for an additional 1 minute until fragrant.
Sprinkle the flour over the vegetables and stir continuously for 2 minutes to form a roux.
Gradually whisk in the chicken broth and milk, stirring constantly to avoid lumps.
Cook the sauce over medium heat, stirring frequently, until it thickens and bubbles, about 5-7 minutes.
Remove the sauce from heat and stir in salt, black pepper, dried thyme, and dried parsley.
In a large mixing bowl, combine the cooked chicken, cooked rice, steamed broccoli, and the prepared sauce. Mix well to combine.
Stir in 2 cups of shredded cheddar cheese and 1 cup of shredded mozzarella cheese into the mixture.
Transfer the mixture into two greased 9×13-inch casserole dishes, dividing evenly.
In a small bowl, combine the crushed buttery crackers, grated Parmesan cheese, remaining 1 cup cheddar cheese, and 1 cup mozzarella cheese.
Sprinkle the cracker and cheese topping evenly over both casseroles.
Bake uncovered in the preheated oven for 45 minutes, or until the casserole is bubbly and the topping is golden brown.
Let the casseroles rest for 10 minutes before serving.