Sprinkle a small amount of water on a large plastic sheet of plastic wrap. Place 2 of the breast halves on top of the plastic and sprinkle again with water. Cover with another sheet of plastic wrap and pound with a rolling pin or meat pounder until about 1/4-inch thick. Repeat with the remaining 2 breast halves.

Mix the flour with the salt and pepper in a shallow pie plate. Heat half of the olive oil in a large skillet over medium-high heat until almost smoking. Working in 2 batches, place the chicken in the flour mixture and turn to coat on all sides. Shake off the excess flour and add to the skillet. Cook until lightly browned and cooked through, 3 to 4 minutes per side. Transfer to a plate or platter and cover loosely with foil. Repeat with the remaining oil and chicken.

Pour off the fat from the skillet and return the skillet to the heat. Add the chicken stock and lemon juice. Increase the heat to high and bring to a boil, stirring to pick up and browned bits in the bottom of the pan. Simmer for 3 minutes. Return the chicken to the skillet and simmer, turning often, until warmed through and the sauce is thickened, about 2 minutes. Add the butter and the capers. Season with salt and pepper and heat just until the butter has melted. Serve on warmed plates with a spoonful of the skillet sauce, topped with lemon slices and parsley.