Preheat the oven to 375 degrees F.
Heat a medium-large skillet over high heat. Add the olive oil. Season the chicken all over with salt and pepper, and lay the pieces in the pan rounded-side down first (the side that had the skin). Cook the chicken, turning only once, until golden brown on both sides, about 5 minutes. Transfer chicken to an oven-proof dish, reserving the skillet. Place the chicken to the oven and bake until an instant-read thermometer inserted into the thickest part of the chicken registers 170 to 75 degrees F, about 15 minutes.
Meanwhile, pour the vermouth into the reserved skillet, scraping up any browned bits that cling to the pan with a wooden spoon, and cook over medium heat until the liquid reduces by about half. Add the chicken broth and continue to cook until slightly thickened.
In a small bowl whisk a teaspoon or so of cold water with the cornstarch, then whisk the mixture into the broth and bring to a full boil. Remove from heat and add the zest, lemon juice, and tarragon. Season with salt and pepper, to taste.
Remove the chicken from the oven. Pour any pan drippings from the chicken into the sauce and stir to combine. Divide the chicken pieces between 4 plates. Pour the sauce over the chicken and serve.
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