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Place flour in a shallow dish. Season chicken with salt and pepper, then coat with flour, shaking off excess.
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In a large skillet, heat oil and 1 tablespoon butter over medium-high. In batches, cook chicken until browned and cooked through, about 3 minutes per side. Transfer to a plate and tent loosely with foil.
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Reduce heat to medium, add thyme, mushrooms, and remaining 2 tablespoons butter, and cook until softened, 6 minutes.
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Add wine and broth and cook, stirring, until reduced by half, 3 minutes. Season with salt and pepper. Return chicken to pan along with any accumulated juices; top with parsley.