Rinse turkey well and pat dry with paper towels. Season inside cavity generously with salt and pepper. Rub one of the cut garlic cloves on inside cavity. Place garlic cloves, onion, and the first apple in cavity.
Pull turkey skin to back; fasten with skewer. If a band of skin crosses the tail, tuck drumsticks under band. If no band, tie drumsticks securely to tail. Twist wing tips under back. Brush turkey with 2 tablespoons melted butter. Season with salt and pepper.
Place bird, breast side up, on a rack in a shallow roasting pan. Insert a meat thermometer into center of one of inside thigh muscles, but not touching bone. Cover bird loosely with foil. Roast in a 325 degree oven about 2-1/2 to 3-1/2 hours or until thermometer registers 160 degrees F. (Cut band of skin or string between drumsticks after 2-1/2 hours.)
Toss apple wedges with lemon juice. Place apples around turkey. Continue to roast, covered with foil, for 30 minutes more.
Meanwhile, in a small saucepan bring apple cider and cinnamon sticks to boiling. Reduce heat and boil steadily about 30 minutes or until cider is reduced to 2/3 cup. Add 1/3 cup butter, brown sugar, and thyme. Heat and stir until sugar is dissolved. Remove and discard cinnamon sticks.
Remove foil from turkey; brush turkey and drizzle apples with the cider mixture. Continue roasting, uncovered, until meat thermometer registers 180 degrees F, brushing bird and drizzling cider mixture over fruit every 10 minutes. Remove turkey from oven; discard cavity ingredients. Transfer roasted bird to a large serving platter. Surround with apples. Cover with foil; let stand 15 to 20 minutes before carving turkey.
Pour pan drippings into a large measuring cup. Scrape the browned bits from the pan into the cup. Skim and reserve fat from the drippings. Pour 1/4 cup of the fat into a medium saucepan (discard remaining fat). Stir in flour. Add enough broth to remaining drippings to equal 2 1/2 cups; add broth mixture all at once to flour mixture in saucepan. Cook and stir over medium heat until thickened and bubbly. Cook and stir for 1 minute more. Season to taste with salt and black pepper.