Adjust an oven rack to the lower-middle position, set a large heatproof plate on the rack, and heat the oven to 200 degrees.
Halve one lemon pole to pole. Trim the ends from one half and cut it crosswise into slices 1/8 to 1/4 inch thick; set aside. Juice the remaining half and whole lemon to obtain 1/4 cup juice; reserve.
Sprinkle both sides of the cutlets generously with salt and pepper. Measure the flour into a shallow baking dish or pie plate. Working with one cutlet at a time, coat with the flour and shake to remove the excess.
Heat 2 tablespoons oil in a heavy-bottomed 12-inch skillet over medium-high heat until shimmering. Lay half of the chicken cutlets in the skillet. Sauté the cutlets until lightly browned on the first side, 2 to 2 1/2 minutes. Turn the cutlets and cook until the second side is lightly browned, 2 to 2 1/2 minutes longer. Remove the pan from the heat and transfer the cutlets to the plate in the oven. Add the remaining 2 tablespoons oil to the now-empty skillet and heat until shimmering. Add the remaining chicken cutlets and repeat.
Add the shallot or garlic to the now-empty skillet and return the skillet to medium heat. Sauté until fragrant, about 30 seconds for shallot or 10 seconds for garlic. Add the stock and lemon slices, increase the heat to high, and scrape the pan bottom with a wooden spoon or spatula to loosen the browned bits. Simmer until the liquid reduces to about 1/3 cup, about 4 minutes. Add the lemon juice and capers and simmer until the sauce reduces again to 1/3 cup, about 1 minute. Remove the pan from the heat and swirl in the butter until the butter melts and thickens the sauce. Stir in the parsley and season with salt and pepper to taste. Spoon the sauce over the chicken and serve immediately.