01 - Slice each chicken breast horizontally to make 4 thin cutlets. Pat dry thoroughly with paper towels and season both sides generously with salt and pepper.
02 - Lightly coat each cutlet in flour, shaking off any excess. Set aside on a clean plate.
03 - Heat olive oil and 2 tablespoons butter in a large skillet over medium-high heat until shimmering. Add chicken in a single layer, cooking in batches if necessary to avoid overcrowding. Sear for 2-3 minutes per side until golden brown and cooked through. Transfer to a plate and cover loosely with foil to keep warm.
04 - Pour wine into the hot skillet, using a wooden spoon to scrape up any browned bits from the bottom. Let simmer for 2 minutes to allow alcohol to cook off slightly.
05 - Add chicken broth, lemon juice, and capers to the skillet. Simmer for 3-4 minutes until liquid reduces slightly and flavors meld together.
06 - Reduce heat to low. Swirl in remaining 2 tablespoons butter, stirring constantly until melted and sauce becomes glossy and emulsified.
07 - Return chicken and any accumulated juices to the skillet. Simmer for 1-2 minutes just until heated through. Plate the chicken, spoon sauce generously over each piece, and garnish with fresh parsley.