Preheat the oven to 350 degrees F. Butter a 9-by-13-inch baking dish.
In a large skillet set over medium-high heat, melt the butter. Add the celery and onions and cook, stirring, until fragrant and softened, about 5 minutes. Stir in the parsley, sage, thyme, salt and pepper and cook for 1 more minute. Add the chicken broth and scrape up any brown bits that have formed on the bottom of the skillet, cooking for an additional minute. Remove from the heat and cool slightly.
In a large bowl, combine the cooled mixture with the stuffing and beaten eggs and mix well. Transfer the mixture to the prepared baking dish.
Bake until golden brown on top, 30 to 35 minutes. Let cool 5 to 10 minutes. Top with additional parsley and serve.