To cook the chicken, season with salt and pepper. Heat the butter and oil over medium heat in a large, straight-sided skillet or saucepan (not cast iron, as it will interact with the acidic tomatoes and vinegar). Add the chicken, skin side down, without crowding the pan. Cook without crowding until a rich, golden brown on both sides, 3 to 5 minutes per side. Remove the chicken to a plate.
Pour off all but 1 Tbsp. of fat. Add the garlic and cook until fragrant, 45 to 60 seconds. Add the vinegar and scrape with a wooden spoon to loosen any brown bits from the bottom of the skillet. Stand back when adding the vinegar, because when it hits the hot pan it will produce strong vinegar fumes. Return the chicken to the skillet. Add the tomatoes with their liquid, chicken stock, bay leaf, parsley, and thyme. Bring to a boil, then decrease the heat to medium-low, cover, and cook until the chicken is tender and the juices run clear, an additional 30 to 45 minutes. Taste and adjust for seasoning with salt and pepper.
To make the sauce, remove the chicken pieces to a warm serving dish. Increase the heat to high, and cook the sauce until reduced and lightly thickened, about 5 minutes. Taste and adjust for seasoning with salt and pepper. Remove and discard the herbs. Spoon the sauce over the chicken and serve.