Whisk together the lemon juice, 1/4 cup of the olive oil, the shallot and 1/4 teaspoon black pepper in a large baking dish.
Pound each chicken breast between sheets of waxed paper or plastic wrap with a rolling pin or a mallet to 1/8-inch thickness. Add the chicken to the baking dish, turn to coat, cover and marinate in the refrigerator for at least 30 minutes and up to 4 hours.
Heat your grill to high.
Remove the chicken from the marinade and season on both sides with salt. Grill for 2 to 3 minutes per side or until golden brown and just cooked through.
While the chicken is grilling, combine the arugula, tomatoes, and onion, if using, in a large bowl. Toss with the vinegar and remaining 2 tablespoons of the oil and season with salt and pepper.
Place each chicken cutlet on a large plate and divide the arugula-tomato salad on top. Top the salad with a few shavings of cheese and a drizzle of olive oil.