Prepare the Chicken: Pat the chicken thighs dry with paper towels and season generously with salt and pepper on both sides.
Sear the Chicken: Heat the olive oil in a large skillet over medium-high heat. Once the oil is shimmering, place the chicken thighs skin-side down. Cook for 5-7 minutes until the skin is golden and crispy. Flip the chicken and cook the other side for 3-5 minutes. Remove the chicken from the skillet and set aside on a plate.
Sauté Aromatics: Lower the heat to medium. Add the shallot and garlic to the skillet, stirring frequently for 1-2 minutes until fragrant and softened.
Deglaze the Pan: Pour in the vinegar, scraping up any browned bits from the bottom of the pan. Let the vinegar reduce slightly for about 1 minute.
Create the Sauce: Add the chicken stock, cream, and Dijon mustard to the pan, stirring to combine. Bring the mixture to a gentle simmer.
Return Chicken to the Pan: Nestle the chicken thighs back into the skillet, skin-side up. Cover the skillet with a lid and cook for 15-20 minutes, until the chicken is fully cooked (internal temperature of 165°F/74°C).
Finish the Sauce: Remove the chicken from the skillet and keep warm. Stir the chopped tarragon and optional butter into the sauce, allowing it to thicken slightly for 2-3 minutes.
Serve: Plate the chicken and generously spoon the vinegar pan sauce over each piece. Garnish with additional fresh tarragon if desired.