Cut the chicken breasts in half lengthwise to create 6 cutlets. Sprinkle both sides with salt and pepper.
In a shallow dish, add 1/2 cup flour. Coat the chicken cutlets with flour, shaking off any excess, and place them on a large plate in a single layer.
Heat a large skillet over medium-high heat. Add 2 Tbsp olive oil and 2 Tbsp butter. Once the butter melts and sizzles, add 3 chicken cutlets and sauté for 2-4 minutes per side or until they reach 165°F on an instant-read thermometer. Remove to a serving platter. Repeat with remaining chicken, adding 1 Tbsp oil and 1 Tbsp butter to the skillet.
In the same skillet, add 1/2 cup wine, 1 Tbsp butter, 1/4 cup lemon juice, and 1/4 cup rinsed capers without reducing the heat. Simmer for 1 minute. Dip the chicken pieces into the sauce a few at a time and return them to the platter. Continue simmering the sauce for another 2 minutes to slightly thicken. Drizzle the sauce over the chicken and garnish with lemon slices and fresh parsley.