Preheat oven to 400 degrees F.
To make the stuffing, melt butter in a skillet over medium heat. Place apple pieces in skillet and sauté. Add diced onions and sauté until softened, about 3 minutes. Remove from heat. Stir in tarragon, breadcrumbs, apple cider, salt, and pepper. Set aside.
Loosen skin on breasts and thighs of chicken by sliding fingers under the skin. Push stuffing under skin, distributing evenly. Tie legs with a six-inch piece of twine. Sprinkle 1 teaspoon salt and 3/4 teaspoon black pepper on the inside and outside of the chicken. Place onion half inside the cavity of the chicken. Place chicken in roasting pan; bake for 1 hour at 400 degrees.
To make the sauce, transfer chicken from the roasting pan onto a cutting board. Scrape bits from the bottom of the pan, then pour pan drippings into a saucepan. 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 and arrange on serving plate. Sauce may be served on the side for individual glazing. Makes 4 servings.
