PreparationCook chicken: Put oven rack in middle position and preheat oven to 350°F. Pat chicken dry and sprinkle with 1 teaspoon salt and 1/2 teaspoon pepper. Heat oil and 1 tablespoon butter in a 10- to 12-inch heavy skillet over moderately high heat until foam subsides, then brown chicken in 2 batches, turning over once, 8 to 10 minutes per batch. Transfer chicken as browned to a small (13- by 9-inch) flameproof roasting pan, arranging it in 1 layer. Once all chicken is browned, roast, uncovered, in oven 15 minutes. Make sauce while chicken roasts: Discard all but 2 tablespoons fat from skillet, then add garlic, shallots, carrot, thyme sprigs (2), and bay leaves and cook over moderately high heat, stirring and scraping up brown bits, until vegetables are browned, 3 to 6 minutes. Add tomato paste and cook, stirring, 1 minute. Add wine, vinegar, sugar, and remaining teaspoon salt and 1/2 teaspoon pepper and boil, stirring occasionally, until reduced by half, 5 to 10 minutes. Finish sauce and chicken: Add chicken broth to sauce along with any juices from roasting pan and bring to a boil. Pour through a fine-mesh sieve onto chicken in roasting pan, pressing on and then discarding solids. Continue to roast chicken, uncovered, until just cooked through, about 15 minutes more. Transfer chicken with tongs to an ovenproof serving dish and keep warm, uncovered, in turned-off oven. Meanwhile, boil sauce in roasting pan over high heat, stirring occasionally, until reduced to a scant cup, then remove sauce from heat and swirl in remaining 2 tablespoons butter. Season with salt and pepper and pour sauce over chicken.