Preheat the oven to 375 degrees and butter a 3-quart baking dish. Melt 2 tbsp butter in a large skillet over medium heat. Add the sausage and cook, stirring until browned, about 5 minutes. Remove with a slotted spoon and transfer to a plate.
Melt the remaining 6 tbsp butter in the skillet, then add the onion, celery, sage, 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 potato bread, cornbread, sausage, and vegetable-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 for 30 minutes, then uncover and bake until golden, about 35 more minutes.