Pound chicken breasts with a meat mallet to 1/2-in. thickness; sprinkle with salt and pepper. In a large skillet, heat oil over medium heat; cook chicken until no longer pink, 5-6 minutes per side. Remove from pan; keep warm.
Add mushrooms and onion to pan; cook and stir over medium-high heat until tender and lightly browned, 2-3 minutes. Add garlic; cook and stir 30 seconds. Add wine; bring to a boil, stirring to loosen browned bits from pan. Cook until liquid is slightly reduced, 1-2 minutes; serve over chicken.
Nutrition Facts 1 chicken breast half with 1/4 cup mushroom mixture: 243 calories, 7g fat (2g saturated fat), 94mg cholesterol, 381mg sodium, 5g carbohydrate (2g sugars, 1g fibre), 36g protein.