Preheat the oven to 350°F (175°C).
In a large skillet, heat olive oil over medium heat. Add the chopped onion and minced garlic and sauté until softened, about 3-4 minutes.
In a large mixing bowl, combine the cooked shredded chicken, cooked rice, thawed peas, sautéed onion and garlic, cream of mushroom soup, sour cream, milk, salt, black pepper, and dried thyme. Mix well until all ingredients are evenly incorporated.
Stir in 1/2 cup of the shredded cheddar cheese and 1/2 cup of the shredded mozzarella cheese into the mixture.
Transfer the mixture into a greased 9×13-inch baking dish and spread evenly.
Sprinkle the remaining 1/2 cup cheddar cheese and 1/2 cup mozzarella cheese evenly over the top.
Sprinkle breadcrumbs evenly over the cheese layer to add a crispy topping.
Bake uncovered in the preheated oven for 35-40 minutes, or until the casserole is bubbly and the top is golden brown.
Remove from oven and let it rest for 5 minutes before serving.