Melt butter in heavy medium skillet over medium-high heat.
Sprinkle chicken with salt and pepper.
Add to skillet and cook until golden, about 4 minutes per side.
Transfer chicken to plate.
Add shallots to skillet and sauté 30 seconds.
Add vinegar; boil until reduced to glaze, about 2 minutes. Stir in broth. Return chicken, skin side up, to skillet. Reduce heat to medium-low, cover and simmer until cooked through, about 12 minutes. Using tongs, transfer chicken to 2 plates.
Add tarragon to liquid in skillet. Increase heat to medium-high; boil uncovered until liquid is slightly reduced, about 2 minutes. Spoon sauce over chicken.