Preheat the oven to 350°F (175°C).
In a large skillet, heat olive oil over medium heat. Add chopped onion and minced garlic and sauté for 3-4 minutes until softened.
Add the chicken pieces to the skillet and cook for 5-6 minutes until they are no longer pink on the outside. Remove from heat.
In a large mixing bowl, combine the uncooked rice, chicken broth, cream of mushroom soup, sour cream, milk, dried thyme, salt, and black pepper. Stir until well combined.
Add the cooked chicken, sautéed onions and garlic, and frozen peas to the rice mixture. Stir to evenly distribute all ingredients.
Transfer the mixture into a greased 9×13-inch baking dish and spread evenly.
Cover the baking dish tightly with aluminum foil and bake in the preheated oven for 40 minutes.
Remove the foil and sprinkle the shredded cheddar and mozzarella cheeses evenly over the top of the casserole.
Return the casserole to the oven uncovered and bake for an additional 10 minutes, or until the cheese is melted and bubbly.
Remove from the oven and let the casserole rest for 5 minutes before serving.