Preheat the oven to 350°F (175°C). Lightly grease a 9×13 inch casserole dish.
In a medium saucepan, bring chicken broth to a boil. Add the uncooked white rice, reduce heat to low, cover, and simmer for 15 minutes or until rice is tender and liquid is absorbed.
While the rice cooks, steam the broccoli florets until just tender, about 4-5 minutes. Drain and set aside.
In a large mixing bowl, combine sour cream, cream of mushroom soup, mayonnaise, chopped onion, minced garlic, salt, black pepper, and paprika. Stir until smooth.
Add the cooked chicken, steamed broccoli, cooked rice, and 1 cup of shredded cheddar cheese to the bowl. Mix everything together until well combined.
Transfer the mixture into the prepared casserole dish and spread evenly.
Sprinkle the remaining 1/2 cup cheddar cheese and 1 cup mozzarella cheese evenly over the top.
In a small bowl, mix the breadcrumbs with melted butter until evenly coated. Sprinkle the breadcrumb mixture over the cheese layer.
Bake uncovered for 30-35 minutes, or until the casserole is bubbly and the breadcrumb topping is golden brown.
Remove from the oven and let it rest for 5 minutes before serving.