1. In a large, deep frying pan, heat the oil over moderately high heat. Season the chicken with 1/4 teaspoon each of the salt and pepper and add to the pan. Cook until browned, turning, about 8 minutes in all. Remove. Pour off all but 1 tablespoon fat from the pan.
2. Add the butter to the pan and reduce the heat to moderately low. Add the onion and cook, stirring occasionally, until translucent, about 5 minutes. Raise the heat to moderately high. Add the mushrooms, garlic, and 1/4 teaspoon of the salt. Cook, stirring frequently, until the vegetables are browned, about 5 minutes.
3. Add the flour and cook, stirring, for 30 seconds. Stir in the vermouth and bring back to a simmer. Stir in the broth, tomatoes, thyme, and the remaining 1/2 teaspoon salt. Add the chicken and any accumulated juices. Reduce the heat; simmer, covered, until the chicken is done, about 10 minutes. Stir in the parsley and the remaining 1/4 teaspoon pepper.
Wine Recommendation: This earthy dish is perfectly suited to the rustic charms of a country red wine from southwestern France. Look for a bottle from one of the various appellations in that region, such as Cahors, Madiran, or Bergerac.