In a separate small mixing bowl, whisk together eggs.
Pat chicken dry with a paper towel and season generously with salt and pepper. Using a meat pounder, gently pound chicken until it is 1/4 inch thick. Place on a cutting board, and slice chicken breasts in half.
Dip slicedchicken breasts into the egg, then dredge in flour mixture, dip back into the egg mixture, and into the flour one more time. Dipping the chicken in the egg and flour mixtures twice helps create an extra thick and crispy crust. Repeat until all chicken is coated.
In a large skillet, heat oil until hot and shimmering.
Place chicken in the skillet and let fry for 5-8 minutes on each side. Cook until chicken is crispy and golden-brown, but juicy in the center.
Toss arugula and tomatoes with oil, lemon, salt, and pepper. Add-in parmesan if using.
Place a bed of arugula on a plate and add chicken. Top with a handful of tomatoes and arugula. Serve warm with a few wedges of lemon.
*Cooking time will vary based on thickness of chicken. Make sure to keep an on eye on them to avoid overcooking. Make sure to add more oil if needed during the cooking process.