Slice the chicken breasts in half horizontally to create 4 thin cutlets. If they are thicker than ½-inch or uneven in thickness, pound them with a meat mallet or rolling pin to an even thickness, no thicker than ½-inch. (Cover the chicken with plastic wrap, parchment paper, or place it in a zip-top bag before pounding to prevent splattering.)

Season both sides of the chicken cutlets with salt and pepper.

Place the flour in a shallow bowl or baking dish. Dredge each piece of chicken in the flour, shaking 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 necessary, brown the chicken on both sides, about 4-5 minutes per side. The chicken should be nearly cooked through but not fully done. Transfer to a clean plate. (If browning the chicken in two batches, add an extra 1-2 tablespoons of olive oil as needed.)

Add the lemon juice, white wine (or chicken broth), and capers to the skillet. Stir, scraping up any browned bits from the bottom of the pan, and cook for 2 minutes. Taste and adjust seasoning with salt and pepper if needed.

Return the chicken to the pan and simmer in the sauce until the internal temperature reaches 165°F. Transfer to a serving plate.

Add the remaining 1 tablespoon of butter to the skillet, whisking for 1 minute. Remove from heat and drizzle the sauce over the chicken. Garnish with chopped fresh parsley and lemon slices. Serve immediately.