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In a nonstick skillet, heat 1 teaspoon of the oil.
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In a bowl, mix 2 tablespoons of the vinegar, the mustard and garlic. Add the chicken and coat both sides with the mixture.
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Transfer the chicken and mixture to the skillet; saute until cooked through, about 3 minutes per side.
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Transfer to a plate and keep warm.
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In the skillet, heat the remaining teaspoon of oil. Saute the mushrooms about a minute; add the broth/wine, thyme and remaining tablespoon of vinegar. Cook stirring until the mushrooms are deep brown, about 2 minutes longer.
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Serve the chicken, topped with the mushrooms.