Slice each chicken breast horizontally to make 4 cutlets. Place a cutlet between two sheets of parchment paper or plastic wrap and gently pound with a meat mallet or rolling pin to about 1/4-inch thickness; repeat with remaining cutlets.
Season both sides of each pounded cutlet with kosher salt and freshly ground black pepper.
In a shallow bowl, beat the large egg with 1 teaspoon of water.
In another shallow bowl, combine the seasoned whole wheat breadcrumbs and grated Parmesan.
Dip each seasoned cutlet into the beaten egg, then press into the breadcrumb mixture to coat both sides. Place coated cutlets on a work surface and spray both sides lightly with olive oil spray.
Preheat the air fryer to 400°F (200°C).
Working in batches as needed, arrange cutlets in a single layer in the air fryer basket. Air fry at 400°F for 7 minutes, flip, and cook until golden and cooked through, about 6–8 more minutes per batch (total about 12–15 minutes per batch) depending on thickness.
Divide baby arugula among plates (about 1 1/2 cups arugula per plate) and top with a cooked chicken cutlet. Squeeze lemon wedges over the cutlets and arugula before serving.