Slice each chicken breast in half horizontally to create 4 thin cutlets. If thicker than 1/2 inch or uneven, place inside a zip-top bag and pound gently with a meat mallet or rolling pin to an even 1/2-inch thickness.

Season both sides of each chicken cutlet generously with salt and black pepper.

Pour the flour into a shallow dish. Dredge each cutlet in the flour and shake off any excess.

Heat the olive oil and 2 tablespoons of the butter in a large skillet over medium-high heat. Working in batches if needed, cook the chicken 4 to 5 minutes per side until golden brown and almost cooked through. Transfer to a clean plate.

Add the lemon juice, white wine (or chicken broth), and capers to the skillet. Cook for about 2 minutes, scraping up the browned bits from the bottom of the pan. Taste and adjust salt and pepper.

Return the chicken to the skillet and cook until the internal temperature reaches 165 degrees F, about 2 to 3 more minutes. Transfer the chicken to a serving plate.

Add the remaining 1 tablespoon of butter to the skillet and whisk for 1 minute off the heat until the sauce is glossy. Spoon the sauce over the chicken. Garnish with chopped fresh parsley and lemon slices and serve immediately.