Position a rack on the lowest level in the oven that will let you fit the turkey and still have room to bake the dressing. Heat the oven to 350ºF. Have ready a large roasting pan with a roasting rack, preferably V-shaped, set in the pan.
Prepare the turkey: Remove the bag of giblets from the main and neck cavities of the bird and discard the liver. Set aside the neck, tail, gizzard, and heart. Rinse the turkey well and pat it dry with paper towels.
Season the chest cavity with some of the salt and a few grinds of pepper. Put four of the onion quarters, the garlic, the apples, four of the thyme sprigs, and four of the sage leaves in the chest cavity. Secure the legs under the flap of tail skin or tie them together. Bend the wingtips back behind the turkey, securing any loose neck skin underneath the wingtips.
Roast the turkey: Brush the turkey with the butter. Season the turkey with the rest of the salt and more pepper. Put the turkey, breast side down, on the roasting rack. Scatter the remaining onion quarters and herbs in the roasting pan. Add the neck, tail, gizzard, and heart to the pan, along with 1 1/2 cups of the apple cider and 1 cup of the broth (you'll use the remaining cider for basting and the remaining chicken broth for the gravy). Roast for 30 minutes. Baste the turkey with the pan juices, and roast another 30 minutes.
Remove the turkey from the oven. Using oven mitts covered with aluminum foil, or wads of paper towels, turn the turkey breast side up. (The turkey won't be very hot at this point.) Baste with the pan juices, return the turkey to the oven with the breast facing the opposite direction from the way it was (you will have flipped and rotated it). Roast for another 1 hour, basting occasionally.
After the turkey has roasted for 2 hours total, begin basting with the remaining apple cider every 30 minutes, rotating the position of the pan to brown the bird evenly. The turkey is done when an instant-read thermometer registers 170ºF when inserted into the thickest part of the thigh. (When the internal temperature of a thigh is 155º to 160°F, the turkey is about 30 minutes away from being done; a 14-pound unstuffed turkey takes about 3 hours.)
When the turkey is done, tilt it over the roasting pan to let the juices in the main cavity run into the pan. Transfer the turkey to a carving board or serving platter and tent it loosely with foil. Let it rest for 20 to 30 minutes before carving.
While the turkey rests, make the gravy: Discard the solids in the roasting pan and skim off as much fat as possible with a large spoon (tilting the pan makes this easier). Set the roasting pan over medium-high heat and bring the juices to a simmer. Using a wooden spoon, scrape and loosen any browned bits sticking to the bottom and sides of the pan. In a measuring cup, mix the flour with the remaining 1 cup chicken broth. Add the flour mixture to the pan and whisk until thickened, about 3 minutes. If the gravy gets too thick, thin it with more chicken broth until it reaches the consistency you like. Continue cooking and stirring for another 5 minutes to eliminate any raw flour flavor. Season to taste with salt and pepper. Strain into a gravy boat or a small bowl; keep warm until ready to serve.