Preheat your oven to 350°F (175°C).
In a large mixing bowl, combine the stuffing mix with the melted butter and chicken broth. Stir well and let it sit for a few minutes to absorb the liquid.
In a large skillet over medium heat, combine the shredded rotisserie chicken, cream of chicken soup, cream of mushroom soup (if using), sour cream, milk, salt, pepper, and dried thyme. Stir until well combined.
Add the frozen mixed vegetables, if using, and stir to combine.
Spread the chicken and vegetable mixture evenly in a 9×13-inch baking dish.
Spread the prepared stuffing mix evenly over the top of the chicken mixture.
Cover the baking dish with aluminum foil and bake in the preheated oven for 20 minutes. Remove the foil and bake for an additional 10-15 minutes, or until the top is golden brown and the casserole is heated through.
Allow the casserole to cool slightly before serving.