Adjust oven rack to upper-middle position and heat oven to 450 degrees. Dissolve cornstarch in chicken broth. Whisk in vinegar, honey and set aside.
Pat chicken dry with paper towels and season with salt and pepper. Heat oil in 12" skillet over medium-high heat until just smoking. Cook chicken skin side down until well browned, about 6-8 minutes. Transfer to plate, skin side up.
Pour off all but 1 T fat from skillet and return to medium-high heat. Add shallot and garlic and cook until fragrant, about 30 seconds. Whisk broth mixture and add to skillet, scraping up any browned bits. Bring to a boil and return chicken to skillet skin side up, along with accumulated juices. Move skillet to oven and cook until breasts register 160 degrees and drumsticks/thighs register 175 degrees, about 10-15 minutes.
Transfer chicken to serving platter, tent loosely with foil and let rest while preparing sauce. Return skillet to medium-high heat (skillet handle will be hot) bring to boil and cook until sauce is slightly thickened 5-7 minutes. off heat, whisk butter, tarragon and any accumulated juices from platter into sauce. Season with salt and pepper to taste. Spoon sauce over chicken and serve.