Spray the air fryer basket with oil or line with parchment. Place the chicken halves between plastic wrap and pound to 1/4-inch thickness. Season both sides with garlic powder, salt, and pepper.
In a shallow bowl, whisk the egg, 2 tablespoons lemon juice, and 1/2 teaspoon garlic powder. In a second bowl, combine the flour, Parmesan, parsley, and 1/2 teaspoon salt.
Dip each cutlet in the egg mixture, then press into the flour mixture to coat both sides. Arrange in a single layer in the basket and spritz the tops with oil.
Air fry at 370°F for 12 to 15 minutes, flipping and spritzing the second side halfway through, until golden and the internal temperature reaches 165°F. Repeat with any remaining cutlets.
While the chicken cooks, melt the butter in a small saucepan over medium-low heat. Add the broth, 3 tablespoons lemon juice, and capers. Simmer 3 to 4 minutes until slightly reduced.
Spoon the sauce over the chicken and serve with lemon slices and parsley.