Surround the chicken pieces with buttered noodles that have been sprinkled with freshly chopped parsley, or leave the bird whole and serve with rice.
Rinse and dry chicken. Put salt, pepper, 1 teaspoon tarragon, 3 tablespoons butter, and a little of the vermouth in the cavity of the chicken. Truss chicken. Brown the chicken on all sides in the remaining butter in a heavy cast-iron pot. Remove from pot. Add carrot, onion, and remaining tarragon, and cook for a few minutes. Add remaining vermouth or wine and return chicken to the pan on its side. Cover with a tight-fitting lid and cook at a bare simmer. During the cooking time (about 1 hour total) use a wooden spoon to rotate the chicken to its breast, then the other side, and finally breast-side up (cooking about 15 minutes on each side). When the chicken is done (prick second joint to see if juices run clear), remove it to a cutting board to rest. Strain and degrease the liquid in the pan, and add chicken broth. Mix the cornstarch with some of the liquid and whisk it into the pot. Stir, bring liquid to a boil, and correct seasonings. Cut the chicken into serving pieces with poultry shears and arrange on a platter. Spoon a little of the sauce over the chicken and serve the rest in a gravy boat.