Take each chicken breast and slice lengthwise. Place each slice of chicken breast between two pieces of wax paper and pound with the flat end of a meat mallet until very thin.
Place the arugula in a large bowl. Add the lemon juice and 2 tablespoons of the extra-virgin olive oil, season with salt and pepper and toss.
Heat the remaining tablespoon of olive oil in a large skillet, preferably cast iron, over medium-high heat.
Season the chicken breasts generously with salt and pepper, add to the skillet and cook, turning once, until browned on both sides and just cooked through, about 2-3 minutes for each side. You may have to work in batches, and add a little more olive oil between batches.
Transfer the chicken to warm serving plates, and top each portion with some of the arugula salad. Using a vegetable peeler shave some of the parmigiano-reggiano cheese over the arugula and serve.
