Prepare the chicken by slicing each breast horizontally to create two thin cutlets. Place between plastic wrap and pound gently until 1/4 inch thick.
Season both sides of the chicken cutlets with salt, pepper, and garlic powder. Dredge lightly in flour, shaking off any excess.
Heat 2 tablespoons olive oil and 2 tablespoons butter in a large skillet over medium high heat. Once the butter stops foaming, add chicken cutlets in a single layer.
Cook chicken for 3 to 4 minutes per side until golden brown and cooked through. Internal temperature should reach 165 degrees F. Transfer to a warm platter.
Pour out excess fat from the skillet. Add remaining 1 tablespoon olive oil. Pour in white wine and scrape up browned bits from the bottom of the pan.
Simmer wine for 2 minutes to reduce slightly. Add chicken broth and lemon juice, bring to a simmer and cook for 3 to 4 minutes until reduced by one third.
Reduce heat to low. Stir in capers and lemon slices. Simmer for 1 minute to warm through.
Remove from heat. Whisk in remaining 2 tablespoons cold butter, one piece at a time, until melted and glossy.
Taste and adjust seasoning with additional salt or lemon juice. Stir in lemon zest and half the parsley.
Return chicken to skillet briefly to warm through and coat with sauce, or plate and spoon sauce over top. Garnish with remaining parsley and serve immediately.