Preheat oven to 200°F and place a sheet pan inside to keep cooked chicken warm.
Slice chicken breasts horizontally to make thin cutlets; optionally cut into smaller pieces.
Pat chicken dry, season with salt and pepper, then dredge lightly in flour, shaking off excess.
Heat 1/4 cup olive oil in a large cast iron skillet over medium-high heat until shimmering.
Sear chicken in batches for about 3 minutes per side until golden brown; transfer to warm oven.
In the same skillet, add a drizzle more olive oil and melt ghee or butter over medium heat.
Stir in lemon juice, white wine (or chicken broth), and capers; bring sauce to a boil while scraping browned bits.
Return chicken to skillet, reduce heat, and simmer for 4 to 5 minutes, basting chicken with sauce.
Remove from heat, garnish with chopped parsley, and serve with lemon wedges and sauce drizzled on top.