Preheat oven to 400 degrees.
Melt butter in a large skillet over medium heat. Saute the celery and onion until soft and translucent, 10 to12 minutes.
In a large bowl, add the celery, onion, cornbread, stuffing mix, and sage. Stir in the chicken stock and eggs and mix until completely combined. Season with a few grinds of black pepper. The mixture should be fairly soupy. Pour into a lightly greased 9 by 13-inch baking dish. Bake, uncovered, until the dressing is set and golden brown, 45 minutes to an hour.
Cook's Note: The skillet cornbread can be made up to 2 days in advance. The dressing can be assembled the day before and baked on the day of serving.
Preheat the oven to 425 degrees. Grease a large cast-iron skillet and place in the oven to heat while mixing the batter.
In a large bowl, mix the cornmeal, flour, sugar, baking powder, salt, baking soda, and a few generous grinds of black pepper.
In a separate bowl, whisk the buttermilk, eggs, and melted butter together until combined. Pour into the dry ingredients and stir until just combined (it's okay if it's a little lumpy). Pour the batter into the hot cast-iron skillet and bake until the top is golden brown, about 25 minutes.
Immediately remove the cornbread from the skillet and allow to cool. The cornbread can be made up to 2 days in advance when using for the dressing. Store until needed, then crumble and follow instructions for the dressing.
If enjoying the cornbread immediately (not for the dressing recipe), remove the cornbread from the oven, invert it onto a large plate, and then flip it once more so it is right side up. Slice into wedges and serve warm with lots of butter.