Rinse the chicken and pat dry. Slice and pound the chicken into cutlets (see Creating Cutlets below). Season generously with salt and pepper to taste.

Place the flour in a shallow bowl, season with salt and pepper and stir to mix.

Heat 2 tablespoons of the oil and 1 tablespoon of the butter in a large skillet over medium-high heat until sizzling hot. Dredge both sides of each piece of chicken in the flour, pressing gently to help the flour adhere and shaking off any excess before placing in the skillet.

Sauté until golden brown around the edges, working in batches and careful not to overcrowd the skillet, about 3 minutes per side. Remove, place on a platter and cover loosely with foil to keep warm. Repeat with the remaining chicken, adding more oil, 1 tablespoon at a time, if needed. If the flour in the bottom of the skillet starts to get too dark, clean the skillet halfway through by removing it from the heat and carefully wiping the skillet dry with paper towels, but keep a little flour in the bottom of the skillet, as it will help your sauce thicken.

Add 1 tablespoon of oil to the same skillet and heat until sizzling hot. Add the garlic, capers and lemon slices and cook, stirring occasionally, about 1 minute. Add the wine, scraping up any brown bits from the bottom of the pan. Bring to a boil and cook until the wine reduces by half, about 3 minutes. Add the broth and lemon juice and continue to cook, stirring occasionally, until the sauce thickens. Reduce the heat to low and stir in the remaining butter to melt.

Add the chicken and any juices that have collected in the bottom of the platter back to the skillet with the sauce and heat through, about 3 minutes. Place the chicken back on the platter with the sauce poured over. Sprinkle with parsley and serve warm.