Thaw turkey if frozen. In a medium saucepan combine water, 1/2 cup honey, and about half of the parsley, sage, thyme, and garlic. Bring to a boil. Remove from heat. Stir in olive oil. Cover; let stand 30 minutes. Strain through a fine-mesh sieve; discard solids.
Rinse turkey; pat dry with paper towels. Skewer the neck skin to the back. Tie legs to tail. Twist wing tips under the back.
Place the turkey, breast side up, on a rack in a shallow roasting pan. Using a flavor-injector syringe, inject the honey mixture into the meat of the turkey. (This may take up to 20 injections, so try to evenly distribute the seasoned mixture in the turkey. If the syringe gets clogged with a bit of seasoning, you will need to flush it out with water and a toothpick.) Sprinkle turkey lightly with salt and pepper.
Insert a meat thermometer into the center of one of the inside thigh muscles without the thermometer touching the bone. Cover turkey loosely with foil. Roast in a 325 degree F oven for 2-1/2 hours.
For glaze, in a small saucepan, heat and stir 1/4 cup honey and butter until butter is melted. Stir in remaining parsley, sage, thyme, garlic, salt, and pepper.
Remove turkey from oven. Cut the string between the drumsticks so the thighs will cook evenly. Remove the foil to let the bird brown. Brush about one-third of the glaze over turkey. Return turkey to oven and roast for 30 to 60 minutes more, brushing two more times with remaining glaze, or until meat thermometer registers 180 degrees F and juices run clear.
In a medium saucepan, cook onion in butter until tender. Stir in flour, broth, apple cider, honey and cider vinegar. Cook and stir until thickened and bubbly. Cook and stir 1 minute more. Season to taste with salt and pepper.