Preheat the oven to 350°F (175°C).
Bring a pot of water to a boil and blanch the broccoli florets for 2-3 minutes until bright green and slightly tender. Drain and set aside.
In a large skillet, heat olive oil over medium heat. Add the chopped onion and cook for 3-4 minutes until softened.
Add the minced garlic to the skillet and cook for 1 minute until fragrant. Remove from heat.
In a large mixing bowl, combine the cream of mushroom soup, sour cream, milk, salt, black pepper, dried thyme, and paprika. Stir until smooth.
Add the cooked rice, cooked chicken, blanched broccoli, sautéed onion and garlic mixture, and 1 cup shredded cheddar cheese to the bowl. Mix well to combine all ingredients evenly.
Transfer the mixture to a greased 9×13 inch casserole dish and spread it out evenly.
Sprinkle the shredded mozzarella cheese evenly over the top of the casserole.
Bake in the preheated oven for 30-35 minutes, until the casserole is bubbly and the cheese on top is melted and golden brown.
Remove from oven and let the casserole rest for 5 minutes before serving.