Step 1
Preheat the oven to 350°. In a large enameled cast-iron casserole, heat the olive oil. Season the chicken thighs with salt and pepper and add them to the casserole, skin side down. Cook over moderately high heat, in batches if necessary, turning once, until golden brown, about 12 minutes. Transfer the chicken to a platter.
Step 2
Spoon off all but 2 tablespoons of the fat in the casserole. Add the carrots, garlic and leek and cook over low heat until crisp-tender, 5 minutes. Add the flour and stir for 1 minute. Add the vinegar and stir, scraping up any browned bits from the bottom of the pot. Bring the sauce to a boil and cook until thickened, 3 minutes. Add the broth, season with salt and pepper and bring to a boil. Nestle the chicken in the sauce, skin side up. Transfer the casserole to the oven and braise the chicken for about 50 minutes, until cooked through.
Step 3
Preheat the broiler. Transfer the chicken to a baking sheet, skin side up. Broil on the middle rack of the oven until the skin is golden and crisp, about 4 minutes.
Step 4
Simmer the sauce over moderate heat until reduced to about 4 cups, 10 minutes. Stir in the butter until melted. Add the parsley and chives and season the sauce with salt and pepper. Return the chicken to the casserole, skin side up, and serve with homemade buttered linguini noodles