Preheat the oven to 350°F (175°C).
Steam the broccoli florets until just tender, about 4-5 minutes, then drain and set aside.
In a large mixing bowl, combine the sour cream, mayonnaise, and condensed cream of mushroom soup. Stir in the minced garlic, chopped onion, salt, black pepper, dried thyme, and paprika until well mixed.
Add the cooked chicken and steamed broccoli to the sauce mixture. Stir gently to combine all ingredients evenly.
Transfer the mixture into a greased 9×13 inch casserole dish and spread evenly.
Sprinkle the shredded cheddar cheese, mozzarella cheese, and grated Parmesan cheese evenly over the top of the casserole.
In a small saucepan, melt the butter over medium heat. Remove from heat and stir in the crushed Ritz crackers until coated with butter.
Sprinkle the buttered cracker crumbs evenly over the cheese layer on the casserole.
Bake uncovered in the preheated oven for 30-35 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.