Slice each chicken breast horizontally to make 4 cutlets total, then place each cutlet between two sheets of parchment paper or plastic wrap and gently pound to about 1/4 inch thick.
Season both sides of the pounded cutlets with the kosher salt and freshly ground black pepper.
In a shallow bowl, beat the large egg with 1 teaspoon of water.
In another shallow dish, combine the seasoned whole wheat breadcrumbs and grated Parmesan cheese.
Dip each seasoned cutlet first into the beaten egg, letting excess drip off, then press into the breadcrumb mixture to coat both sides; place the breaded cutlets on a work surface and lightly spray both sides with olive oil spray.
Preheat the air fryer to 400°F (204°C).
Working in batches if needed, arrange cutlets in a single layer in the air fryer basket and cook for 7 minutes, flip each cutlet, then cook an additional 5–8 minutes until golden and the internal temperature reaches 165°F (74°C).
Remove the cutlets from the air fryer and let rest 1–2 minutes. Plate each serving with about 1 1/2 cups baby arugula and squeeze lemon wedges over the chicken to taste.