Cut chicken breasts in half lengthwise so you end up with 6 cutlets. Sprinkle both sides of cutlets with salt and pepper.
Add ½ cup flour to a shallow dish. Dredge chicken to coat with flour, shaking off the excess. Transfer to a large plate in a single layer.
Place a large skillet (non-stick or stainless steel will work) over medium/high heat. Add 2 Tbsp olive oil and 2 Tbsp butter. Once butter melts and begins to sizzle, add 3 pieces of dredged chicken and saute 2-4 minutes per side or until 165˚F on an instant-read thermometer (timing will depend on thickness of chicken). Remove to a serving platter, add additional 1 Tbsp oil and 1 Tbsp butter and sauté remaining chicken.
In the same skillet without reducing heat, add ½ cup wine, 1 Tbsp butter, ¼ cup lemon juice, and ¼ cup rinsed capers. Simmer 1 minute then dip your chicken pieces into your sauce a few pieces at a time and place them back on your platter. Continue simmering sauce another 2 minutes to slightly thicken. Drizzle sauce over chicken and garnish with lemon slices and fresh parsley.
