Preheat oven to 325 degrees F. In a food processor combine cilantro leaves, parsley leaves, 1 Tbsp. melted butter, shallot, lime peel, salt, and pepper. Cover; process with on/off pulses until a chunky paste forms. Divide mixture in half. Set aside.
Remove and discard neck and giblets from turkey. Rinse turkey; pat dry with paper towels. Loosen the skin from the breast meat by sliding your fingers underneath it, being careful not to tear it. Slide your hand as far as you can toward the opposite end of the turkey, separating the skin from the meat. Rub half of the herb mixture underneath the skin over the entire breast, working toward the thighs as much as possible.
Sprinkle inside of body cavity with kosher salt and pepper. Fill cavity with cilantro sprigs, carrots, and onion. Pull neck skin to back and fasten with a small skewer. Tuck drumstick ends under band of skin across the tail, if available. If there is not a band of skin, tie drumsticks securely to the tail using kitchen string. Twist wing tips under back. Place turkey, breast side up, on a rack in a shallow roasting pan. Brush melted butter over turkey. Insert an oven-going meat thermometer into center of inside thigh muscles. The thermometer should not touch bone. Cover loosely with foil.
Roast for 2 3/4 hours. Meanwhile, in a small saucepan melt 1/4 cup butter. Stir in honey and remaining cilantro mixture. Set aside. Remove foil from turkey. Roast for 15 to 45 minutes more or until thermometer reaches 175 degrees F, brushing occasionally with the honey mixture. Remove from oven.
Stir broth into pan drippings from roasted turkey in roasting pan, scraping up any browned bits from bottom of pan. Strain and pour drippings into a 2-cup glass measure. Skim and reserve fat from drippings. If necessary, add enough melted butter to the reserved fat to make 1/4 cup. Add enough broth to the drippings in measuring cup to make 2 cups total liquid.
Pour the 1/4 cup fat into a medium saucepan (discard any remaining fat). Stir in flour. Cook and stir over medium heat for 1 minute. Add broth mixture all at once to flour mixture in saucepan, stirring until smooth. Cook and stir over medium heat until bubbly. Cook and stir for 1 minute more. Season to taste with salt and pepper. Makes 2 cups.