In a small saucepan, bring the maple syrup to a boil. Lower the heat and simmer until reduced by half, about 3 minutes. Pour into a bowl, then stir in 1 stick butter and the chopped thyme. Freeze for 30 minutes.
Position a rack in the lower third of the oven; preheat to 400 degrees . Place the turkey, giblets and neck on a rack in a large, heavy roasting pan. Season with salt and pepper. Using your fingers, gently loosen the skin from the breast and spread about half of the maple butter over the breast meat under the skin. Rub the remaining maple butter on the outside.
In a large bowl, combine the onions, celery, thyme sprigs and salt and pepper to taste; fill the turkey cavity with two handfuls of the onion mixture. Tie the drumsticks together with kitchen string; tuck in the wings. Scatter the remaining onion mixture around the turkey and pour 1 cup water into the pan.
Place the turkey in the oven and lower the temperature to 350 degrees . Roast, basting with the pan juices and 1 cup cider, until an instant-read thermometer inserted in the thickest part of the thigh without touching the bone registers 165 degrees , about 2 hours and 40 minutes (during cooking, cover loosely with foil to prevent overbrowning). Transfer the turkey to a platter and let rest.
Strain the pan drippings into a large glass measuring cup; discard the solids and skim off any fat. Add enough chicken broth to make 3-1/2 cups of liquid. Set aside.
In a small bowl, combine the flour and remaining 1/2 stick butter. Place the roasting pan over 2 burners, add the remaining 1-1/2 cups cider and cook over medium-high heat, scraping up the browned bits. Lower the heat to medium and cook, stirring, for 4 minutes. Slowly whisk in the butter-flour paste and the reserved pan juices. Cook, whisking, until thickened, about 10 minutes. Season with salt and pepper. Serve the gravy with the turkey.