To make the brining solution, dissolve the salt and sugar in 2 gallons of cold water in a non-reactive container (such as a clean bucket or large stockpot, about 16-quart,or a clean, heavy-duty, plastic garbage bag). Add the remaining ingredients. Rinse turkey inside and out and place turkey in container with solution. Gather tops of bags together or cover. Refrigerate turkey in brine 18 to 20 hours. Before moving on to the next stops, remove turkey from brine. Drain very well; discard brine. Pat turkey dry inside and out.
Mix first 6 ingredients in small bowl. Rinse turkey inside and out; pat dry. Place on rack set in large roasting pan. Rub inside of main cavity and neck cavity with orange quarters (reserve orange quarters); sprinkle with salt and pepper. Rub herb butter over outside of turkey.
Set rack at lowest position in oven and preheat to 3500F. If not stuffing turkey, fill main cavity and neck cavity with orange quarters, onion, garlic, celery, thyme sprigs, and sage sprigs. If stuffing turkey, spoon stuffing loosely into both cavities. Tuck wing tips under; tie legs together loosely to hold shape. Mix orange juice and Dijon mustard in small bowl; pour into pan around turkey. Place turkey in oven with tail at back. Roast turkey 45 minutes. Add 2 cups broth to pan; baste turkey. Turn pan around. Roast turkey 45 minutes. Add 1 cup broth; baste with pan juices. Turn pan around and cover turkey loosely with foil. Continue to roast until thermometer inserted into thickest part of thigh registers 1750F, basting with 1 cup broth and rotating pan every 45 minutes, about 2 hours 30 minutes longer if unstuffed and 3 hours longer if stuffed. Transfer turkey to platter; let stand 30 minutes (internal temperature will increase 5 to 10 degrees).Strain pan juices into 8-cup measuring cup. Spoon off fat. Melt butter in large saucepan over medium-high heat. Add onion; sauti 3 minutes. Add flour and stir until golden, about 5 minutes. Whisk in apple and all remaining ingredients, then pan juices. Bring gravy to boil. Reduce heat to medium and simmer until gravy thickens, whisking occasionally, about 10 minutes. Season gravy with salt and pepper.