In a medium bowl, mix 2 tablespoons of vinegar, mustard, and garlic. Add chicken and turn to coat.
Transfer chicken and marinade to skillet. Saute chicken until cooked through, about 3 minutes on each side. Transfer chicken to a platter and keep warm.
Heat remaining teaspoon of oil of skillet. Saute mushrooms for one minute. Add broth, thyme, and remaining tablespoon of vinegar. Cook, stirring occasionally, until mushrooms are deep brown, about 2 minutes.
Serve chicken topped with mushrooms.
*If your chicken breasts are very thick, cut them in half lengthwise. They cook quicker and turn out much juicier if they are thinner.