Cut each chicken breast in half lengthwise and sandwich between two pieces of parchment paper. Use a mallet to gently pound them so they are even and about ½” (12.7mm) thick. Be careful that the meat does not break.
To bread the chicken, set up three separate bowls. Place the flour in the first bowl and season with salt and pepper. In the second bowl beat the eggs. In the third bowl add the breadcrumbs.
Take a piece of chicken and dredge through the flour, then the egg mixture, and then the breadcrumbs. Press the chicken into the breadcrumb mixture so that it is lightly coated on both sides.
Arrange the breaded chicken on a baking sheet and chill for 15 minutes to help the breading adhere to the chicken (optional).
Add enough vegetable oil to a large frying pan so that it is ¼” (6.4mm) deep. Heat until the surface of the oil is shiny. Add one of the breaded chicken pieces to the hot oil and pan-fry until golden brown, 3 minutes. Use tongs to flip the chicken and continue to cook until golden brown and cooked through, about an additional 3 to 4 minutes. Transfer to a baking sheet and repeat with the remaining chicken.
Toss the arugula in olive oil and lemon juice and season with salt and pepper.
Add the arugula salad over the chicken Milanese and garnish with Parmesan and serve with lemon wedges. Serve immediately.