Make the panko dredge: to a shallow bowl, mix unseasoned panko, grated Parmesan cheese, 1 tsp salt and 1/2 tsp pepper.
Use a thin, sharp knife to slice each chicken breast in half lengthwise to create two thin fillets from each chicken breast. Place the chicken breasts between two pieces of parchment paper on a cutting board, and gently pound to about ⅜-inch with a meat tenderizer. Season with remaining salt and pepper.
Whisk the eggs in another wide, shallow bowl. Dip the chicken cutlets in the eggs, then coat with the panko mixture. Place on a platter. Line a baking sheet with paper towels. Preheat oven to warm (170℉-200℉).
Heat 2-3 tablespoons of oil in a large skillet over medium-high heat until the oil is hot and shimmering. Carefully add the chicken. Cook until golden brown, turning after 3 minutes and adjusting the heat as needed so the chicken cooks steadily without over-browning, about 6-7 minutes total. May need to do this in batches.
Using a slotted spatula, transfer to the paper towel lined baking sheet and transfer to oven to keep warm. Repeat with the remaining 2-3 tablespoons of oil and chicken.
Make the arugula salad: Toss the arugula, tomatoes and lemon juice together in a medium bowl and season with salt and pepper. For each serving, place 2 chicken pieces on each plate, and top with salad. Garnish with shaved Parmesan.