Preheat oven to 450 degrees F.
Debone the chicken. Leave the first wing bone on the chicken breasts so that you have a piece called the supreme.
Wrap the dark meat from the legs and thighs and freeze it for another use. Set supreme in fridge and start stock.
Chop the bones and put them in a small roasting pan. Roast the bones for 1 hour, shaking the pan periodically to turn the bones. Remove the bones from the oven.
Reduce oven temperature to 425 degrees F.
Put bones in a stockpot.
Pour off the fat from the roasting pan and discard. Deglaze the roasting pan with white wine. Pour the wine and 3 cups chicken stock over the bones. Bring to a boil. Lower heat immediately to medium low and simmer for 25 minutes. Skim the liquid periodically with a ladle.
Strain the sauce into a fresh pan and let it reduce over medium high heat for 10 minutes or until slightly thickened. Keep stock warm as you cook the chicken supreme.
Mash 3 tablespoons butter with 2 teaspoons tarragon. Loosen the skin of the chicken breasts. Smear half the butter under each breast.
Season the chicken with kosher salt and pepper.
Heat 1 tablespoon clarified butter in an oven-proof sauté pan until smoking.
Add the chicken breasts, skin side down and brown for 4 minutes. Shake the pan once or twice to keep the skin from sticking.
Put the pan in the oven and cook for 4 minutes at 425 degrees F. Flip the breasts over and continue cooking in the oven about 4 to 5 minutes longer. Remove chicken to a plate.
Discard all but 2 teaspoons fat from the sauté pan. Deglaze pan with balsamic vinegar. Add the reduced stock and reduce for another 5 minutes. Swirl in the remaining tablespoon of butter. Add 1 tablespoon chopped tarragon and 3/4 teaspoon kosher salt. Serve sauce with chicken. Garnish with tarragon sprigs.