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.