Squeeze the juice from 1 ½ of the lemons and reserve. Lay the remaining lemon half flat side down and cut into very thin slices with a paring knife. Remove the pits and set the lemon slices aside.

Season the scaloppine with salt and pepper. Dredge in flour to coat both sides lightly and tap off excess flour. Heat 3 tablespoons of the olive oil and 2 tablespoons of the butter in a wide, heavy skillet over medium heat until the butter is foaming. Add as many of the scaloppine as will fit without touching and cook until golden brown on the underside, about 3 minutes. Flip and cook until the second side is lightly browned, about 2 minutes. Remove and drain on paper towels. Repeat with remaining scaloppine.

Remove all scaloppine from the pan. Pour off the fat and carefully wipe out the skillet with a wad of paper towels. Pour in the remaining 3 tablespoons olive oil and add the remaining 4 tablespoons butter, the garlic and lemon slices. Cook, scraping the bottom of the skillet, until the garlic is golden brown, about 3 minutes. Scoop out the lemon slices and set aside. Scatter the olives and capers into the skillet and cook, stirring gently, until they begin to sizzle, about 4 minutes. Pour in the wine, bring to a vigorous boil and cook until the wine is reduced in volume by half. Pour in the chicken stock, bring to a boil and cook until slightly syrupy, about 4 minutes. Return the scaloppine to the skillet, turning the cutlets in the sauce until they are warmed through and coated with sauce. Swirl in the parsley and divide the scaloppine among warm plates. Spoon the sauce over them, including some of the capers and olives in each spoonful. Decorate the tops of the scaloppine with the reserved lemon slices.