Pound chicken breasts to 1/2-inch thickness using a meat mallet.
Season both sides of the cutlets with freshly ground black pepper.
Set up two shallow bowls: egg whites in one and breadcrumbs in the other.
Dip each cutlet in egg whites, then coat thoroughly with breadcrumbs.
Spray both sides of the breaded cutlets generously with olive oil spray.
Preheat the air fryer to 400°F for 3 minutes.
Place cutlets in a single layer in the air fryer basket without touching.
Air fry for 12 to 14 minutes, until golden brown and internal temperature reaches 165°F.
Meanwhile, melt butter in a small saucepan over medium heat.
Add lemon juice, chicken broth, and capers; stir to combine.
Simmer the sauce for 3 to 4 minutes until slightly reduced and glossy.
Remove chicken from the air fryer and let rest for 2 minutes.
Pour the lemon-caper sauce over the cutlets and garnish with parsley and lemon slices before serving.