Preheat the oven to 350°F (175°C).
Heat olive oil in a medium skillet over medium heat. Add the chopped onion and minced garlic and sauté until the onion is translucent, about 3-4 minutes. Remove from heat.
In a large mixing bowl, combine the shredded cooked chicken, cooked white rice, thawed peas, sautéed onion and garlic, cream of mushroom soup, sour cream, milk, dried thyme, salt, and black pepper. Stir until well mixed.
Add 3/4 cup of shredded cheddar cheese and 3/4 cup of shredded mozzarella cheese to the mixture and stir gently to combine.
Transfer the mixture into a greased 9×13-inch casserole dish and spread evenly.
Sprinkle the remaining 1/4 cup cheddar cheese, 1/4 cup mozzarella cheese, and grated Parmesan cheese evenly over the top of the casserole.
Bake uncovered in the preheated oven for 35-40 minutes, or until the casserole is bubbly and the cheese on top is melted and golden brown.
Remove from the oven and let the casserole rest for 5 minutes before serving.