Discard any chestnuts that feel light for their size; they are apt to be spoiled or molded. Cut an X about 1/2 inch long through flat side of each shell. Place nuts in a 10- by 15-inch pan.
Bake in a 400° oven until nuts are no longer starchy-tasting (cut 1 open to test), 25 to 30 minutes. Wrap hot nuts in a towel and enclose in a plastic bag; let stand about 15 minutes. Remove 1 warm nut at a time and use a short-bladed knife to pull off shell and as much brown skin as possible; discard shell and skin. Coarsely chop or crumble chestnuts.
Meanwhile, in a 350° oven, toast polenta cornbread cubes in a 12- by 17-inch roasting pan, stirring occasionally, until edges begin to brown, 30 to 40 minutes.
In a 10- to 12-inch frying pan over medium-high heat, turn bacon occasionally until browned, 4 to 6 minutes; lift from pan and drain on towels. When cool, crumble and add to toasted cornbread.
To drippings in frying pan, add onion, celery, and green onions; stir occasionally until limp, 5 to 7 minutes. Add sage, thyme, nutmeg, and chestnuts; stir occasionally until vegetables are lightly browned, about 5 minutes longer. Scrape mixture into pan with cornbread.
Add poultry broth and chopped poultry-broth meats and mix gently; for a moister dressing, add 1/2 cup chicken broth. Season dressing with salt and pepper to taste. Spoon into a shallow 3-quart casserole and cover tightly.
Bake in a 350° oven until hot in the center (about 150°), 35 to 40 minutes. For a crusty top, uncover dressing for the last 20 to 25 minutes.
Bake the polenta cornbread (a variation of the yeast cornbread Luciano makes) or use your favorite cornbread recipe or mix. Instead of roasting the chestnuts, you can use 2 cups peeled cooked chestnuts (canned steamed, water-packed, or vacuum-packed). If making dressing up to 1 day ahead, chill airtight, then bake 50 to 65 minutes.