Preheat oven to 375°.
Crumble the baked cornbread into a large mixing bowl and set aside.
Melt 1/4 cup butter in a large skillet over medium- high heat.
Add the onion, bell pepper, and celery, cook, stirring often until vegetables are softened. Remove from heat.
Stir in the chicken broth, cream of mushroom soup, and beaten egg to the skillet with the cooked vegetables.
Mix until everything is well combined.
Pour the skillet mixture over the crumbled cornbread in the mixing bowl.
Stir in the granulated garlic, sage, salt, and black pepper, mixing well.
Butter a 9×9-inch casserole dish and spread the dressing mixture evenly into the pan.
Top with the shredded chicken and sprinkle with dry rub for extra flavor.
Cover the dish with foil and bake at 375°F for 15 minutes.
Uncover and bake for another 15-20 minutes, until the top is golden brown and set.
Let the dressing rest for a few minutes before serving.