In a medium bowl mix two tablespoons vinegar, the mustard and garlic. Add chicken and turn to coat.
In a non-stick skillet, heat 1 teaspoon of oil.
Transfer chicken and marinade to skillet. Saute chicken until cooked through, about 3 minutes on each side. Transfer chicken to platter and keep warm.
Heat remaining teaspoon of oil in skillet. Saute mushrooms for one minute. Add broth, thyme, and remaining tbsp of vinegar. Cook, stirring occasionally until mushrooms are deep brown, about two minutes longer.