Prepare the Cornbread: Preheat oven to 425°F. Combine self-rising cornmeal mix and, if desired, sugar in a large bowl. Stir together eggs and buttermilk in a medium bowl; add to cornmeal mixture, stirring just until moistened.

Heat salted butter in a 10-inch cast-iron skillet in preheated oven 5 minutes. Stir melted butter into batter. Pour batter into hot skillet.

Bake in preheated oven until Cornbread is golden, about 25 minutes; cool in skillet 20 minutes. Remove from skillet to a wire rack, and cool completely, 20 to 30 more minutes. Crumble Cornbread. Freeze in a large heavy-duty ziplock plastic bag up to 1 month, if desired. Thaw in refrigerator.

Prepare the Dressing: Preheat oven to 350°F. Melt butter in a large skillet over medium-high; add onion and celery, and cook, stirring often, until tender, 10 to 12 minutes. Add sage and thyme, and cook, stirring often, 1 minute.

Stir together eggs in a very large bowl; stir in crumbled Cornbread, onion mixture, stuffing mix, chicken broth, black pepper, and kosher salt until blended.

Spoon mixture into 2 lightly greased 13- x 9-inch (3-quart) baking dishes. Cover and freeze up to 3 months, if desired; thaw in refrigerator 24 hours. (Uncover and let stand at room temperature 30 minutes before baking.)

Bake, uncovered, in preheated oven until lightly browned and cooked through, 1 hour to 1 hour and 15 minutes.