1. The day before serving, prepare the giblet broth (set aside the liver for another use). Place all the broth ingredients in a heavy saucepan and add water to just cover. Bring to a boil, reduce the heat, and simmer until the giblets are tender, skimming any foam that rises to the surface, about 1 hour. Strain the broth over a bowl, reserving the giblets, turkey neck, and chicken backs; discard the vegetables. You should have about 3½ cups of broth.
2. When it is cool enough to handle, shred the meat from the turkey neck and chicken backs, discarding any skin. Finely chop the giblets. Mix the meats and giblets together, cover, and refrigerate until needed.
3. Preheat the oven to 325°F.
4. On serving day, prepare the turkey: Rinse the turkey well inside and out, and pat it dry with paper towels. Remove any excess fat.
5. Squeeze the orange halves inside the body and neck cavities. Sprinkle the cavities with paprika, salt, and pepper. Stuff the cavities loosely with the stuffing, using about 3 cups for the neck and 8 cups for the body. Close with turkey lacers or sew closed with a large needle and heavy thread. Tie the legs together with kitchen string.
6. Rub the turkey all over with the butter and sprinkle with paprika, salt, and pepper.
7. Place the turkey, breast side up, on a rack in a large ovenproof roasting pan. Pour 2 cups of the reserved giblet broth into the bottom of the pan, and cover the turkey loosely with aluminum foil. Cover the remaining broth and refrigerate it. Place the turkey in the oven and roast for 1½ hours.
8. Remove the foil and roast the turkey, basting with the pan juices every 30 minutes, for 2½ hours.
9. Raise the oven temperature to 350°F and roast until an instant-reading thermometer inserted into the thickest part of the thigh reads 180°F, another 1 to 1¼ hours. The temperature in the deepest part of the breast should read 160°F. The juices should run clear when the thigh is pricked with a small sharp knife.
10. Remove the turkey, place it on a platter, and let it rest about 20 minutes, loosely covered with aluminum foil. Reserve the juices in the pan.
11. Remove the stuffing from the body and neck cavities and cover it with aluminum foil to keep warm.
12. Prepare the giblet gravy: Heat the pan juices in the roasting pan over medium heat, scraping up all of the brown bits on the bottom of the pan. Pour the juices through a gravy separator to remove the fat, or skim the fat off with a metal spoon. Pour the defatted juices into a measuring cup.
13. Melt the butter in a heavy saucepan over medium heat. Whisk in the flour and continue whisking until it browns slightly, 2 to 3 minutes. Whisking constantly, slowly pour in 2 cups of the reserved pan juices, and continue whisking until smooth. Bring the gravy to a boil, then reduce the heat to medium low and add the Madeira, thyme, salt, pepper, parsley, and reserved giblet mixture (from Step 2). Simmer, stirring, until the gravy has thickened, about 10 minutes. If you prefer a thinner gravy, add more of the remaining 1½ cups giblet broth until the desired consistency is achieved. Adjust the seasonings. Transfer the gravy to a gravy boat.
14. Present the turkey before carving it. Arrange the carved meat on a large decorative platter. Garnish with the fresh sage and serve with the gravy.