Preheat the oven to 350°F (175°C).
In a large saucepan, bring the chicken broth to a boil. Add the uncooked rice, reduce heat to low, cover, and simmer for 15 minutes until the rice is partially cooked.
While the rice cooks, heat olive oil in a large skillet 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 an additional 1 minute until fragrant.
Add the chicken pieces to the skillet and cook for 5-7 minutes until they are no longer pink in the center.
In a large mixing bowl, combine the partially cooked rice, cooked chicken mixture, frozen peas, cream of mushroom soup, sour cream, dried thyme, salt, black pepper, and paprika. Stir well to combine.
Transfer the mixture into a greased 9×13 inch baking dish and spread evenly.
Sprinkle the shredded cheddar and mozzarella cheeses evenly over the top of the casserole.
Cover the dish with aluminum foil and bake in the preheated oven for 25 minutes.
Remove the foil and bake for an additional 15-20 minutes until the cheese is melted, bubbly, and golden brown.
Remove from the oven and let the casserole rest for 5 minutes before serving.