Dry the cornbread: If your cornbread is fresh, cube it and spread on a baking sheet. Bake at 300°F (150°C) for 15 to 20 minutes, stirring once, until dry and lightly toasted. Let cool.

Brown the sausage: In a large skillet over medium-high heat, cook the sausage, breaking it into small crumbles, for 8 to 10 minutes until browned and cooked through. Remove with a slotted spoon to a large bowl, leaving the fat in the pan.

Cook the vegetables: Add 2 tablespoons of butter to the sausage fat. Add the onion and celery and cook for 6 to 8 minutes until soft. Stir in the sage, thyme, salt, and pepper and cook 1 minute more.

Combine everything: Add the dried cornbread cubes to the bowl with the sausage. Pour the vegetable mixture over the top. Drizzle in the beaten eggs. Pour in 2 1/2 cups of broth and toss gently until the bread is evenly moistened. Add more broth a little at a time if dry spots remain.

Bake: Preheat the oven to 375°F (190°C). Butter a 9x13 baking dish. Spread the dressing in evenly and dot with the remaining 2 tablespoons of butter. Cover with foil and bake 30 minutes. Uncover and bake 20 to 25 minutes more, until the top is golden and crisp.

Rest and serve: Let the dressing rest for 10 minutes before scooping.