Make stuffing by melting butter in non-stick skillet over medium heat. Place apple pieces in skillet; sauté. Add onion diced pieces; sauté until softened, about 3 minutes. Remove from heat. Stir in tarragon, breadcrumbs and apple cider. Add salt and pepper; stir. Set aside. Preheat oven to 400F.
Loosen skin on breasts and thighs of chicken by sliding fingers under skin. Push stuffing under skin, distributing evenly. Tie legs with six inch piece of butcher twine. Sprinkle 1 teaspoon salt and 3/4 teaspoon black pepper on inside and outside of chicken. Place onion half inside the cavity of chicken. Place chicken in roasting pan; bake for 1 hour.
To make sauce, transfer chicken from roasting pan to cutting board. Pour pan drippings into sauce pan, scraping bits from bottom of pan. Whisk in flour. Warm over medium heat. Gradually whisk in apple cider and chicken broth. Bring to simmer; cook over very low heat for 3 minutes. Stir in tarragon and apple cider vinegar.
Carve chicken; arrange on serving plate. Pour sauce into pitcher and pass separately