1. Place chicken between plastic wrap and pound evenly until about 1/4 inch thick.

2. Spread flour in a shallow dish; season with 1 teaspoon salt and 1/4 teaspoon pepper; whisk to combine. Place chicken in seasoned flour, turning to coat thoroughly, then tap off the excess.

3. Heat the oil and 1 tablespoon butter in a large skillet over medium-high heat until butter starts to sizzle. Cook the chicken in batches, if necessary, to avoid crowding the pan (the cutlets should fit snugly in a single layer) until golden and cooked through, 2 to 3 minutes for each side. The cooking time will depend on the thickness of the cutlet. Transfer the chicken to a platter. Pour out any excess fat from the pan.

4. Return pan to medium heat and add wine, scraping up any browned bits from the bottom of the pan with a wooden spoon. Cook until the liquid is reduced by half, about 30 seconds. (If the liquid is reducing too quickly -- before all the browned bits have been incorporated -- remove the pan from the heat.) Remove the pan from the heat. Add lemon juice, capers, and remaining 2 tablespoons butter and swirl until melted and combined, then add parsley and season with salt as desired.

5. Immediately pour the sauce over the chicken, and serve.