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 cornbread 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.