For the salad
Carefully slice each chicken breast in half horizontally. Place each piece of chicken in a large ziplock bag, or between 2 sheets of waxed paper. Using a meat mallet, pound until they are very thin. Season both sides with salt and pepper. Set aside.
In a medium bowl whisk together the milk and eggs. Place the flour in another shallow dish. Season with salt and pepper. Place the breadcrumbs and grated IL Villaggio Piave in a third shallow dish.
Dredge the chicken breast in the flour, then into the egg mixture, and lastly coat both sides in the breadcrumbs. Repeat with remaining breasts.
Heat olive oil in a skillet over medium-low heat. When it’s hot, add 2 pieces of the breaded chicken and cook, flipping once, until the breading is golden brown and the chicken is cooked, 2 to 3 minutes per side; transfer the chicken to a plate. Repeat with remaining chicken.
For the salad, in a large bowl toss the arugula with red onion, cherry tomatoes, lemon juice, olive oil, and shaved IL Villaggio Piave. Season with salt and pepper.
To serve, place 1 piece of chicken on each plate. Top generously with arugula. Serve with a lemon wedge and top with more shaved IL Villaggio Piave, if desired.