Heat a large skillet over high heat with a little drizzle of EVOO. Place chicken breasts between two pieces of plastic wrap. Using the bottom of a heavy pot or small sauté pan, give each chicken breast a good smack to make them even thinner. Peel back the top layer of plastic wrap and drizzle the balsamic vinegar over top and season with a little salt and pepper.
Transfer the chicken to the hot sauté pan, seasoned-side down, and cook on each side for about 2 minutes.
While the chicken is cooking, combine the arugula, with the juice of half a lemon, a drizzle of EVOO and a little bit of salt in a bowl. Remove chicken from the skillet and reserve on a plate while you make the sauce.