Preheat the oven to 375°F (190°C).
In a large oven-safe skillet or casserole dish, heat olive oil over medium heat.
Add the chopped onion and sauté for 3-4 minutes until translucent.
Add the minced garlic and cook for another 1 minute until fragrant.
Add the rinsed rice to the skillet and stir to coat with the oil, onions, and garlic.
Pour in the chicken broth and stir in the dried thyme, dried parsley, salt, and black pepper.
Bring the mixture to a simmer, then remove from heat.
In a separate bowl, combine the shredded chicken, sour cream, cream of chicken soup, frozen peas, and half of the cheddar and mozzarella cheeses. Mix well.
Fold the chicken mixture into the rice mixture evenly.
Sprinkle the remaining cheddar and mozzarella cheeses evenly on top of the casserole.
Cover the skillet or casserole dish tightly with aluminum foil.
Bake in the preheated oven for 35 minutes.
Remove the foil and bake for an additional 10 minutes until the cheese on top is melted and golden.
Remove from oven and let the casserole rest for 5 minutes before serving.