The uniform thickness of scaloppine abets quick, even cooking. To prevent the arugula from wilting, let the chicken cool slightly before serving.
One at a time, place chicken breast halves on a cutting board between two sheets of plastic wrap. Pound with the flat side of a meat mallet or the bottom of a small, heavy skillet until 1/4 inch thick. Sprinkle each piece all over with 1/8 teaspoon salt; season with pepper.
Coat a 12-inch nonstick skillet with cooking spray, and heat over medium heat until hot but not smoking. Put 2 chicken breast halves, smooth sides down, in pan; they should lay flat without touching. Cook 4 minutes; turn, and cook until no longer pink in the center, 2 to 3 minutes more. Wipe skillet clean; repeat with remaining chicken.
Whisk together lemon juice and 4 teaspoons oil in a large bowl. Add arugula; toss to coat. Divide among four plates. Top with a chicken breast half, and drizzle each with 1/2 teaspoon oil. Shave cheese with a vegetable peeler, and divide shavings evenly among the plates. Serve with lemon wedges.