Preheat the oven to 350°F (175°C). Butter a 9 x 13 in (23 x 33 cm) baking dish.
Set a large skillet over medium heat. Brown the sausage, breaking it into small crumbles as it cooks. Transfer the cooked sausage to a large mixing bowl. If necessary, drain excess grease, leaving about 1 tbsp in the skillet.
Add the onion and celery to the skillet. Season with salt and pepper and cook, stirring often, until tender. Add them to the bowl with the sausage.
Add the stuffing mix and toss with the sausage and vegetables. Pour in the chicken broth and melted butter; mix until evenly moistened.
Transfer to the prepared baking dish and spread evenly. Dot the top with a little additional butter.
Cover with foil and bake for 25 minutes. Uncover and bake for about 15 minutes more, until hot and lightly browned on top.