Prepare your dredging station: Whisk eggs, 1 tablespoon lemon juice, and garlic powder together in a bowl until well combined. Place flour, Parmesan, parsley, and salt into a wide, shallow bowl or pie plate; stir with a fork until well blended.

Preheat an air fryer to 375 degrees F (190 degrees C) according to the manufacturer's instructions. Spray the bottom insert of the fryer with cooking spray.

Place chicken between two sheets of heavy plastic on a solid, level surface. Firmly pound chicken with the smooth side of a meat mallet until uniform and thin.

Dredge chicken in flour mixture, then in egg wash, and then in flour mixture again. Place each piece into the air fryer as you go, arranging so no pieces overlap. If you have too many pieces of chicken, you can cook them in batches.

Bake in the preheated air fryer until no longer pink in the center and the juices run clear, about 14 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).

While the chicken cooks, melt butter in a small saucepan over medium heat. Whisk in chicken broth, remaining 2 tablespoons lemon juice, and capers with their liquid. Bring to a simmer, then allow to simmer and reduce slightly while the chicken cooks; the sauce will be thin.

Serve chicken with lemon-butter sauce drizzled over the top.