Prepare the chicken: In a small bowl, stir together 2 teaspoons salt and 1½ teaspoons pepper
Set each chicken breast on a cutting board. Set one palm on top of one breast to hold it still then use a chef’s knife with the other hand to slice the breast in half horizontally. Season generously on both sides with half the salt and pepper.
Put the panko in a shallow bowl, add finely grated Parmesan and olive oil and stir to combine. Coat one cutlet in mayonnaise, then transfer to the panko mixture and turn, pressing panko into it until fully coated. Transfer to a baking sheet and repeat with the remaining chicken. (The breaded chicken will hold in the refrigerator for a few hours.)
Bake on the rack of a foil- or parchment-lined baking sheet at 400 degrees for about 12-15 minutes.
While the chicken cooks, prepare the salad: In a large bowl, whisk to combine the olive oil and lemon juice; season to taste with salt and pepper. Add the arugula, cherry tomatoes (if using) and shaved Parmesan; toss to coat then season to taste again with salt and pepper.
Divide the chicken among plates and pile the salad on the side. Serve immediately, with lemon wedges for squeezing on top.