Preheat the oven to 350°F (175°C).
Bring a pot of water to a boil and blanch the broccoli florets for 2 minutes. Drain and immediately transfer to ice water to stop cooking. Drain again and set aside.
In a medium skillet, heat olive oil over medium heat. Add chopped onion and cook until translucent, about 3-4 minutes. Add minced garlic and cook for an additional 1 minute. Remove from heat.
In a large mixing bowl, combine the cooked rice, cooked chicken, blanched broccoli, sautéed onion and garlic, sour cream, condensed cream of mushroom soup, milk, salt, black pepper, dried thyme, and paprika. Stir until well combined.
Fold in 1/2 cup shredded cheddar cheese and all of the mozzarella cheese.
Transfer the mixture into a greased 9×13-inch baking dish and spread evenly.
Sprinkle the remaining 1/2 cup shredded cheddar cheese evenly over the top.
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 cheese is golden and bubbly.
Remove from the oven and let the casserole rest for 5 minutes before serving.