1. Adjust oven rack to lower-middle position, set large heatproof plate on rack and heat oven to 200.
2. Halve one lemon pole to pole. Trim ends from one half and cut crosswise into slices 1/8 to 1/4 inch thick. Set aside. Juice remaining half and whole lemon to obtain 1/4 cut juice, reserve.
3. Sprinkle both sides of cutlets generously with salt and pepper. Measure flour into pie tin or shallow baking dish. Working one cutlet at a time, coat with flour, and shake to remove excess.
4. Heat heavy bottomed 12-inch skillet over medium-high heat until hot, about 2 minutes; add 2 tablespoons oil and swirl pan to coat. Lay half of chicken pieces in skillet. Saute cutlets, without moving them, until lightly browned on first side, 2 to 2 1/2 minutes. Turn cutlets and cook until second side lightly browned, 2 to 2 1/2 minutes longer. Remove pan from heat and transfer cutlets to plate in oven. Add remaining 2 tablespoons oil to now empty skillet and heat until shimmering. Add remaining chicken pieces and repeat.
5. Add shallot to now empty skillet and return skillet to medium heat. Saute until fragrant, about 30 seconds for shallot. Add stock and lemon slices, increase heat to high and scrape skillet bottom with wooden spoon or spatula to loosen browned bits. Simmer until liquid reduces to about 1/3 cup, about 4 minutes. Add lemon juice and capers and simmer until sauces reduces again to 1/3 cup, about 1 minute. Remove pan from heat and swirl in butter until butter melts and thickens sauce; swirl in parsley. Spoon sauce over chicken and serve immediately.