Pat chicken pieces dry with paper towels and season with 1/2 teaspoon salt and pepper.
Heat 2 teaspoons oil and butter in a large heavy casserole or Dutch oven over medium heat. Add half the chicken pieces and cook, turning occasionally, until browned on all sides, 5 to 7 minutes. Remove to a large plate. Add 2 teaspoons oil to the pot. Add the remaining chicken and cook, turning occasionally, until browned on all sides, 5 to 7 minutes. Remove to the plate.
Heat 2 more teaspoons oil. Add shallots and garlic and cook, stirring, until the shallots are soft and lightly browned, about 1 minute. Add vinegar and bring to a simmer. Return the chicken to the pot. Pour in broth and then carefully nestle the thyme sprigs among the chicken pieces (or stir in dry thyme).
Cover the pot with a tight-fitting lid and simmer over medium-low heat until the chicken is very tender, about 50 minutes.
Just before the chicken is done, whisk sour cream, mustard, tomato paste and flour in a small bowl until smooth. Combine tomatoes, chives, the remaining 1 teaspoon oil and pinch of salt in another small bowl; reserve for
When the chicken is done, remove to a plate, discarding the thyme sprigs (if using). Stir the sour cream mixture into the sauce; bring to a simmer. Reduce heat to low, return the chicken to the sauce and reheat, about 1 minute. Serve garnished with the tomato mixture.