Preheat the oven to 350°F (175°C).
Steam the broccoli florets until tender but still crisp, about 4-5 minutes. Drain and set aside.
In a medium skillet, heat olive oil over medium heat. Add chopped onion and minced garlic, sautéing until softened and fragrant, about 3-4 minutes.
In a large mixing bowl, combine the cooked rice, shredded chicken, steamed broccoli, sautéed onion and garlic, cream of mushroom soup, sour cream, milk, shredded cheddar cheese, salt, black pepper, dried thyme, and paprika. Mix well until all ingredients are evenly incorporated.
Transfer the mixture into a greased 9×13 inch casserole dish and spread evenly.
Sprinkle shredded mozzarella cheese evenly over the top of the casserole.
Bake in the preheated oven for 35-40 minutes, or until the casserole is bubbly and the cheese on top is melted and lightly golden.
Remove from the oven and let it rest for 5 minutes before serving.