In a large deep-sided 12-inch skillet, heat the oil with 1 tablespoon of butter over high heat. Season the chicken thighs well with salt and pepper. When the butter is melted but not smoking, add the chicken and cook on both sides until the skin turns an even golden brown, and the chicken is cooked to the desired doneness, about 12 minutes.
Carefully regulate the heat to avoid scorching the skin. Do not overcrowd the skillet or cook the chicken in several batches.
Transfer the chicken to a serving platter, cover loosely with aluminum foil. Keep warm.
Pour off the fat from the skillet. Return the skillet to medium-high heat and 1 tablespoon of the butter add the wine. Deglaze the pan, scraping up any bits that may have stuck to the skillet. Add the shallots and carrots and cook for 5 minutes without browning them.
Add the tomatoes and cook for several minutes.
Raise the heat to high and slowly add the tarragon vinegar. Cook for another 2 to 3 minutes. whisk in the remaining 1 tablespoons of butter, cook for 1 minute, return the chicken to the skillet, and coat well with the sauce.
Cover and cook over medium heat until the chicken absorbs some of the sauce. Sprinkle with tarragon and turn the chicken pieces to coat. Serve immediately.