Position a rack in the center of the oven and heat the oven to 400°F.
In a small bowl, mix 2 Tbs. of the cider, 1 Tbs. of the mustard, 1 Tbs. of the tarragon, 2 tsp. of the parsley, the butter, 1/2 tsp. salt, and 1/8 tsp. pepper.
Scatter the carrots, fennel, and onion over the bottom of a metal, glass, or ceramic baking dish that measures about 10x15x2 inches. Arrange the chicken pieces, skin side up, on top of the vegetables. Brush the cider-mustard mixture over the chicken pieces and roast for 30 minutes.
Meanwhile, in a small bowl whisk the remaining 1/2 cup cider, 1 Tbs. mustard, the crème fraîche, and cornstarch. Whisk in the chicken broth, vinegar, and 1/2 tsp. salt.
Remove the pan from the oven and reduce the temperature to 375°F. Pour the crème fraîche mixture around the chicken and then scatter the apples around. Return the pan to the oven and roast until the vegetables and apples are tender and an instant-read thermometer registers 165°F in several pieces of chicken, 20 to 30 minutes.
Transfer the chicken to a warmed platter. Use a slotted spoon to arrange the vegetables and apples around the chicken. Sprinkle with a little salt and the remaining 1 tsp. tarragon and 1 tsp. parsley. Tilt the roasting pan so that the juices gather in one corner. With a large, shallow spoon, skim as much fat as possible from the pan sauce. Season the sauce to taste with salt and pepper and pour into a pitcher to pass at the table.