Preheat the oven to 350°F (175°C).
In a large skillet, melt the butter over medium heat. Add the diced onion and celery, cooking until softened, about 5-7 minutes.
In a large bowl, combine the crumbled cornbread, sautéed onion and celery, dried sage, dried thyme, black pepper, and salt. Stir until well mixed.
In a separate bowl, whisk together the chicken broth and beaten eggs. Gradually pour this mixture over the cornbread mixture, stirring gently until combined.
Transfer the mixture to a greased baking dish, spreading it evenly. Bake for 30-35 minutes, or until the top is golden brown and the edges are crispy.
Remove the dressing from the oven and let it cool for 10 minutes before serving.