Preheat the oven to 350°F (175°C).
In a large bowl, combine the shredded cooked chicken, cream of chicken soup, sour cream, milk, garlic powder, onion powder, dried thyme, salt, and black pepper. Mix until well combined.
Add 1 cup of shredded cheddar cheese and 1/2 cup of shredded mozzarella cheese to the chicken mixture. Stir to combine evenly.
In a separate bowl, toss the prepared stuffing mix with the melted butter until the stuffing is evenly coated.
Spread half of the stuffing mixture evenly on the bottom of a 9×13-inch baking dish.
Pour the chicken and cheese mixture over the stuffing layer, spreading it evenly.
Top with the remaining stuffing mixture, spreading it evenly over the chicken layer.
Sprinkle the remaining 1 cup of cheddar cheese and 1/2 cup of mozzarella cheese evenly over the top of the casserole.
Cover the baking dish with aluminum foil and bake for 25 minutes.
Remove the foil and bake for an additional 15 minutes, or until the cheese is melted and bubbly and the top is golden brown.
Remove from the oven and let the casserole rest for 5 minutes before serving.