Set a sheet pan in the oven and preheat to 200°F to keep the chicken warm as you cook in batches.
Place your hand on top of the chicken and carefully slice horizontally to make two thinner cutlets. Optionally, cut them in half crosswise for smaller pieces.
Pat the chicken dry, season with salt and pepper, then coat with flour on both sides, shaking off excess.
In a large cast iron skillet, heat olive oil over medium-high. Cook chicken in batches until golden, about 3 minutes per side. Keep the cooked chicken warm in the oven.
In the same skillet over medium heat, add more olive oil. Stir in ghee or butter, lemon juice, wine, and capers. Bring to a boil, scraping off any browned bits.
Return the chicken to the skillet with the sauce. Simmer for 4 to 5 minutes, basting the chicken. Garnish with parsley. Serve with the sauce drizzled over the top and lemon wedges on the side.
Notes
You can serve this dish with pasta or a side of vegetables for a complete meal.