Heat oil in a large high-sided skillet over medium-high. Sprinkle chicken thighs evenly with pepper and 2 teaspoons salt. Place chicken in skillet, skin side down, and cook, undisturbed, until skin is deep golden brown, 12 to 15 minutes. Flip chicken, and cook for 3 minutes. Transfer chicken to a large plate; set aside. (Chicken will not be cooked through.)

Spoon off and discard all but 2 tablespoons drippings in skillet. Add carrots, onion, and remaining ½ teaspoon salt to skillet; cook over medium, stirring occasionally, until vegetables are softened, 6 to 8 minutes. Add garlic; cook, stirring constantly, until fragrant, about 1 minute.

Add vinegar and mustard to skillet, and stir to scrape up any browned bits from bottom of pan. Bring to a boil over high; cook until liquid is slightly reduced, 2 to 3 minutes. Add broth, and return to a boil. Nestle reserved chicken into skillet, skin side up. Reduce heat to low; simmer, uncovered and undisturbed, until an instant-read thermometer inserted into thickest part of chicken registers 170°F, about 15 minutes. Transfer chicken to a clean plate, and set aside.

Increase heat to medium-high, and cook sauce, stirring occasionally, until slightly reduced, 6 to 8 minutes. Whisk together cornstarch and 1 tablespoon cold water in a small bowl; add to sauce in skillet. Cook, stirring constantly, until sauce is thickened, about 1 minute. Remove from heat. Stir in butter, tarragon, and 1 tablespoon chives; season with additional salt to taste. Nestle reserved chicken into skillet, and garnish with remaining 1 tablespoon chives.