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 for 3-4 minutes until softened and fragrant.
In a large mixing bowl, combine the shredded cooked chicken, sautéed onion and garlic, cream of chicken soup, sour cream, milk, dried thyme, salt, and black pepper. Stir until well mixed.
Add 4 cups of the prepared stuffing mix to the chicken mixture and stir until combined.
Lightly grease a 9×13 inch casserole dish with butter or cooking spray.
Spread the chicken and stuffing mixture evenly into the prepared casserole dish.
Sprinkle the shredded sharp cheddar cheese and shredded mozzarella cheese evenly over the top of the casserole.
Sprinkle the remaining 2 cups of stuffing mix evenly over the cheese layer.
Drizzle the melted butter evenly over the top of the casserole.
Bake in the preheated oven for 35-40 minutes, or until the casserole is bubbly and the top is golden brown.
Remove from the oven and let the casserole rest for 5 minutes before serving.