Preheat oven to 350°
In a large enameled cast-iron pot, heat olive oil.
Season chicken thighs with salt and pepper and add to pot, skin side down.
Cook over moderately high heat, in batches, turning once, until golden brown, about 12 minutes.
Spoon off all but 2 tablespoons of fat in pot.
Add carrots, garlic and onion - cook over low heat until tender - 5 minutes.
Add flour and stir for 1 minute.
Add cider vinegar and stir, scraping up any browned bits from bottom of the pot. Bring sauce to a boil and cook until thickened, 3 minutes.
Add broth, thyme and season with salt and pepper. Bring to a boil.
Nestle the chicken in sauce, skin side up. Transfer pot to oven and braise chicken for about 50 minutes, until cooked through.
Remove pot from oven and preheat to broil.
Transfer chicken to a baking sheet, skin side up. Broil on middle rack of the oven until skin is golden and crisp, about 5 minutes.
Simmer sauce over moderate heat until reduced to about 1/2 - 10 to 15 minutes.
Stir in butter and season with salt and pepper.
Return the chicken to the casserole, skin side up, and serve.