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 3 tablespoons of the butter. Add the sausage and saute until browned, 5 minutes; transfer to a mixing bowl to cool. Melt the remaining 3 tablespoons butter in the same skillet and add the celery, onions and bell peppers. Cook, stirring, until fragrant and softened, about 5 minutes. Stir in the 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 mixing bowl, combine the cornbread stuffing and cooled veggies and sausage. Add the 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 before serving.