If using boneless chicken breasts, slice them in half horizontally to create two thinner halves. Pound these halves, or the pre-cut chicken cutlets using a meat mallet between layers of plastic wrap, or in an open plastic zip-top bag, to about 1/4-inch thickness.
Prepare 3 plates or breading trays. Place flour in the first with 1/2 teaspoon each kosher salt and black pepper. In the second, beat the eggs with the cool water and a pinch of salt and pepper. In the third, mix together panko, Parmigiano Reggiano, and a pinch of salt.
Bread each chicken cutlet by dredging both sides in flour, then dipping in the egg wash to coat, and finally in the panko, gently pressing the breadcrumbs into the cutlet to adhere. In each step, let the excess coating fall off before moving to the next ingredient.
Heat about 1/4 inch of oil or clarified butter in a large skillet over medium high heat until shimmering. Cook cutlets 2-3 at a time (don't crowd the pan), for about 3-4 minutes per side, until the chicken is cooked through (165 degrees F on an instant read thermometer) and the breading is golden.
Transfer chicken cutlets to a platter and finish with a sprinkling of kosher or sea salt and a squeeze of fresh lemon juice. Top with arugula salad and serve.
Spray chicken cutlets on each side with olive oil or vegetable oil spray.
Cook chicken cutlets at 390-400 degrees F, flipping once, until chicken is cooked through and breading is crispy and lightly browned (about 10-15 minutes total). Cooking times will vary by air fryer brand - keep a close eye on your first batch to determine the correct cooking time for the model you're using.