Slice chicken breasts in half lengthwise. Place them on a flat surface, cover with wax paper, and pound to 1/4 inch thickness.
Set up a breading station using 3 shallow bowls. To the first bowl, add the flour. To the second bowl, add the beaten eggs. To the third bowl, add the breadcrumbs, grated Parmesan, Italian seasoning, salt & pepper. Mix well.
Bread the chicken by dipping each breast in flour, then in beaten eggs, and finally in the breadcrumb mixture. Be sure chicken is fully coated after each step.
Heat 2-3 Tablespoons olive oil in a large skillet over medium heat. Add the chicken and pan fry undisturbed for about 3 minutes until first side is golden brown. Flip to the second side, adding additional oil as necessary, and pan fry for an additional 3 minutes. Chicken is done when both sides are golden brown and a thermometer inserted in the center reaches 165°F.
Make the salad by combining arugula, halved tomatoes, shaved Parmesan, olive oil, lemon juice, salt & pepper in a large bowl. Toss gently to mix.
Serve chicken with arugula salad on the side. Garnish with kosher salt, fresh cracked pepper, minced parsley, and additional shaved Parmesan as desired.