To make the turkey: Remove the neck and giblets and set them aside. Mix the salt and rosemary in a small bowl. Sprinkle a small amount of the mixture inside the turkey cavity and the rest over the outside of the bird, moving the legs and wings to reach all areas.
Place the salted turkey in an extra-large zip-top bag (3-gallon size), seal the bag, and chill the turkey for 3 days, turning the bird over once a day.
Remove the turkey from the bag and place it breast-side up on a rack in a roasting pan. Chill it uncovered overnight or up to 24 hours.
Remove the turkey from the refrigerator 1 hour before cooking. Heat the oven to 450ºF.
Turn the turkey breast-side down on the rack in the roasting pan and roast for 30 minutes. Remove it from the oven and use heatproof mitts to carefully turn it breast-side up. Reduce the heat to 325ºF and roast until a thermometer inserted in a thigh (without touching bone) registers 165ºF, 2 to 3 hours longer, depending on the weight of the turkey.
Remove the turkey from the oven and let it rest on the rack in the roasting pan at least 30 minutes or up to 1 hour. Use heatproof mitts to tilt the bird and pour the juices into the roasting pan. Transfer the turkey to a platter. Spoon 3 to 4 cups of hot Chestnut-Mushroom Stuffing (see directions below) into the cavity of the bird if desired. Carve and serve.
To make the stuffing: Heat the oven to 450ºF. Spread the bread cubes in a single layer on rimmed baking sheets and toast, shaking them in the pans once or twice, until light brown, 4 to 6 minutes. (The bread can be toasted just before you put the turkey in the oven.) Transfer the toasted bread to a large bowl.
Reduce the heat to 325ºF. Warm the oil in a large deep skillet over medium-high heat. Add the onions and celery and cook until soft, 5 minutes. Add the garlic and mushrooms and cook until the mushrooms release their liquid and shrink slightly, 5 minutes more. Remove the vegetables from the heat and stir in the nuts, parsley, sage, salt, and pepper. Stir the mixture into the bread cubes along with the broth and eggs.
Coat a 2½- to 3-quart shallow baking dish (a 13-by-9-inch pan works well) with cooking spray or butter. Spread the stuffing in the dish. Bake until heated through and crusty on top, 35 to 45 minutes. (The stuffing can be baked along with the turkey near the end of its cooking time.)
Tip: Note the total weight of the turkey to determine how much salt is needed for seasoning. Use 1 tablespoon of kosher salt for every 5 pounds of turkey (for a 12-pound bird, use 2½ tablespoons of kosher salt; for a 15-pounder, use 3 tablespoons).
You can use frozen and thawed peeled chestnuts or look for the jarred variety in your grocery store’s baking aisle around the holidays.
To make gravy: While the turkey cooks, put the neck and giblets (except the large dark-reddish-brown liver—discard that) into a saucepan with ½ cup of chopped onion,½ cup of chopped celery, 1 bay leaf, and 1 tablespoon of chopped fresh (or 1 teaspoon dried) thyme or parsley. Cover the giblet mixture with 1 inch of water and bring it to a boil over high heat. Reduce the heat to low and cook for 2 hours. Strain out and discard the solids; then pour the liquid into the turkey roasting pan after removing the bird. Place the pan over medium-high heat and bring the liquid to a boil, scraping up all the brown bits. In a cup, stir together 3 tablespoons of cornstarch and ¼ cup of cold water. Add the mixture to the roasting pan and continue boiling the gravy over medium-high heat, stirring constantly, until it thickens. Season with ground black pepper to taste.