Cut the breads into 1-inch cubes and spread them out onto baking sheets. Leave out overnight to get stale. (Alternately, bake them in a 300 degree F oven for 30 minutes to dry them out).
In a large skillet over medium heat, add the butter. When it is melted add the onions, celery, and carrots and season with salt and pepper. Cook until the vegetables are soft, about 10 minutes. Turn off the heat and let cool a bit.
Preheat the oven to 350 degrees F. Butter a 9 by 13-inch baking dish.
Put the bread cubes into a large bowl with the cornbread stuffing and add the cooled vegetables. In a medium bowl, whisk together the broth, gravy, eggs, and poultry seasoning. Pour over the bread and mix; the mixture should be fairly moist but not wet. If it seems to dry, add some more broth. Fill the baking dish with the stuffing and top with the Parmesan. Bake until the top is browned and the stuffing is cooked through, 35 to 40 minutes.