Heat oven to 450º F. In a small bowl, combine the maple syrup and orange juice to make a glaze; set aside.
Place the turkey in a large metal roasting pan, breast-side down. Rub the turkey with half the oil and sprinkle with half the salt and pepper. Turn the turkey over and carefully separate the skin from the breast.
Combine the thyme, sage, and butter and spread it under the skin. Tie the legs together with cooking twine. Rub the surface of the turkey with the remaining oil and sprinkle with the remaining salt and pepper. (The recipe can be prepared to this point up to 1 day ahead; cover and keep refrigerated.)
Place the turkey and the Chestnut-Fig Stuffing in the oven. After 30 minutes, pour the orange-maple glaze over the turkey and cover loosely with aluminum foil. Continue roasting, basting with the pan drippings every 30 minutes. Roast the turkey for 1 1/2 to 2 hours or until a thermometer inserted into the thigh registers 165º F. Remove the stuffing after 1 hour and set aside. Put the Cider-Roasted Vegetables in the oven.
Remove the turkey from the roasting pan and let rest, covered, for 15 minutes. Pour off the excess fat from the pan, add the vermouth, and cook over medium heat on the stovetop until the liquid is reduced by half, about 2 minutes. Add the chicken broth and simmer until thickened, 4 to 5 minutes.
Carve the turkey, serve, and pass the pan gravy.