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If chicken breasts are thicker than 1/2-inch, place them in a large zip top plastic bag or between two sheets of plastic wrap and pound them using a meat mallet until 1/2-inch thick.
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Season chicken with salt and pepper. Dredge in flour and shake off excess.
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Heat olive oil in a deep skillet or sauté pan over medium-high heat.
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Add chicken to the pan and cook until browned on one side, about 3 minutes.
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Turn chicken over and brown the other side, about 3 minutes.
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Remove chicken from pan and place on a plate.
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Lower heat to medium. Add onions and cook, stirring occasionally, until soft and translucent, about 5 minutes.
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Add garlic and cook until fragrant, about 1 minute.
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Return chicken and any juices on the plate to the pan.
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Add mushrooms, broth, vinegar, bay leaf, thyme, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Bring to a boil then reduce to medium-low heat. Cover and cook until chicken is cooked through (has an internal temperature of 165°F when tested with an instant-read thermometer).
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Remove chicken from the pan and place on a warm serving platter. Cover chicken loosely with aluminum foil.
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Bring the sauce to a boil over medium-high heat. Cook for 7 to 10 minutes, until reduced about half.
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Add butter and stir until melted. Discard bay leaf.
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Spoon mushroom sauce over chicken and serve.