Set a sheet pan in the oven and heat the oven to 200°F (this is to keep the chicken warm as you work in batches).

Place your hand on top of the chicken and use a sharp knife to carefully slice horizontally, starting at the thickest side, to make two thinner cutlets. Optionally, cut them in half crosswise to make smaller pieces.

Pat the chicken dry and season all over with salt and pepper. Lightly coat on both sides with the flour and shake off any excess flour.

In a large cast iron skillet, heat a thin layer of olive oil (about ¼ cup) over medium-high until shimmering but not smoking. Working in batches, cook the chicken until golden, about 3 minutes per side. Transfer the seared chicken to the oven to keep warm.

In the same skillet (over medium heat), add a drizzle more olive oil. Add the ghee or butter and stir until melted, then stir in the lemon juice, wine, and capers. Bring the sauce to a boil, scrapping off any browned bits from cooking the chicken earlier.

Return the cooked chicken to the skillet with the sauce. Lower the heat and let simmer for 4 to 5 minutes, basting the top of the chicken with the sauce. Remove from the heat and garnish with the parsley. Serve with the pan sauce drizzled over top and lemon wedges on the side.