Preheat oven to 375°.
In a large skillet over medium-high heat, melt 4 tablespoons of the butter. Add onions, carrots and celery and saute until tender. Transfer to a large mixing bowl and toss with parsley, sage, chestnuts, apples, dried cranberries, mushrooms and bread. Pour eggs, stock and remaining 12 tablespoons of melted butter over, season with salt and pepper and toss again well. If the stuffing is too dry add a bit more stock or water. It should not be saturated or soggy, though.
Place the turkey on a rack in a roasting pan. Rub the cavity opening with salt and pepper. With your fingers, carefully separate as much of the turkey's skin from the breast, as you can, trying not to puncture any holes in the skin as you go.
In a small mixing bowl stir together 8 tablespoons of the softened butter with 1 cup of the brown sugar. Smear about half of the softened butter and sugar mixture over the breast under the skin. Rub the remaining butter and sugar mixture all over the turkey.
Loosely stuff the turkey with the cooled chestnut stuffing. If there is extra stuffing bake it in a buttered baking dish next to the turkey in the oven.
Roast the stuffed turkey about 3 to 3 1/2 hours, basting often during the last hour of cooking with the buttered rum glaze, or until an instant-read thermometer inserted into the thickest part of the turkey leg registers 155°; it will continue cooking for another 20 minutes or so out of the oven. If the turkey skin starts to get too dark, tent it with foil while it roasts.
Butter Rum Glaze: While turkey is cooking, in a small saucepan combine remaining 8 tablespoons of the butter with 1 cup of the brown sugar, ground cinnamon, freshly ground nutmeg, dark rum and a little salt and pepper. Bring to a boil then lower and simmer for 10 minutes until slightly thickened and syrup-like, stirring often.
Transfer the turkey to a carving board to rest and loosely tent with foil and allow to cool. Brush the outside of the turkey with syrup mixture to get a glazed look.