Preheat the oven to 350 degrees F. Generously butter a 13 by 9-inch baking dish and set aside.
In a large skillet, cook the sausage until brown and the fat is rendered, about 5 minutes. Add onions, celery, bell peppers, and garlic, and cook for 2 minutes. Remove from the heat and transfer to a large bowl to cool.
With your fingers, crumble the corn bread into the bowl, add bread, the green onions, parsley, and thyme, and mix well with your hands. Add the salt, pepper, cayenne, and eggs, and mix well with your hands. Add enough broth, 1/2 cup at a time, to moisten the dressing, being careful not to make it mushy.
Transfer to the prepared dish and cover with aluminum foil. Bake until heated through, about 25 minutes. Uncover and bake until golden brown, about 15 minutes.
Preheat the oven to 400 degrees F.
Pour the vegetable oil into a 9-inch baking pan or heavy cast iron skillet. Place the pan into the oven as it preheats, allowing it to heat for at least 10 minutes.
Combine the cornmeal, flour, baking powder, salt, and cayenne in a large mixing bowl and stir with a wooden spoon. Add the buttermilk and egg to the mixture, and stir well to blend. Pour the cornmeal batter into the preheated pan and bake in the oven for 25 minutes or until lightly golden brown. Remove from the oven and let cool before serving or using in the dressing.
When making the dressing, make cornbread the day before.