Preheat your oven to 350°F (175°C). Grease a 9x13-inch baking dish and set aside.
In a large mixing bowl, combine the crumbled cornbread and cubed bread.
In a large skillet, melt the butter over medium heat. Add the chopped onion and celery, sautéing until softened, about 5-7 minutes.
Pour the cooked onion and celery mixture over the bread mixture in the bowl. Stir to combine.
Gradually add the chicken broth, stirring to moisten the bread mixture. The mixture should be moist but not soupy.
Stir in the beaten eggs, dried sage, thyme, salt, pepper, and cayenne pepper (if using). Adjust seasoning to taste.
Pour the dressing mixture into the prepared baking dish and spread it out evenly.
Bake for 35-45 minutes, or until the top is golden brown and the dressing is set in the center.
Remove from the oven and let it cool slightly before serving.