Preheat the oven to 375 degrees F and butter a 3-quart baking dish. Melt 2 tablespoons butter in a large skillet over medium heat. Add the sausage and cook, stirring, until browned and crispy, about 5 minutes. Remove with a slotted spoon and transfer to a plate.
Melt the remaining 6 tablespoons butter in the skillet, then add the onion, celery, thyme, 1 teaspoon salt and a few grinds of pepper. Cook, stirring, until the vegetables are tender, about 5 minutes. Add 3 cups broth and bring to a boil; remove from the heat.
Whisk the eggs and parsley in a large bowl. Add the white or wheat bread, cornbread, sausage, and vegetable and broth mixture and stir until combined, adding up to 1/2 cup more broth if necessary. Transfer to the prepared baking dish. Drizzle with the turkey drippings (or dot with butter).
Cover the dish with foil and bake 30 minutes, then uncover and bake until golden, about 35 more minutes.