Prepare the lemon-caper sauce: Melt butter in a small saucepan over medium heat. Add garlic and cook, stirring occasionally, until aromatic, about 1 minute.

Whisk in flour and cook, whisking constantly, until slightly thickened, about 2 minutes.

Whisk in chicken broth, white wine, and lemon juice and bring to a boil. Remove saucepan from heat and whisk in capers, parsley, lemon zest, salt, and pepper. Cover to keep warm until ready to use and gently whisk before serving.

Preheat an air fryer to 400 degrees F (200 degrees C) according to manufacturer's instructions for 10 minutes. Lightly coat the air fryer basket with nonstick spray.

Cut each chicken breast in half crosswise. Pound each chicken breast half until 1/2-inch thick between sheets of plastic wrap.

Place flour in a medium bowl. Whisk together egg and lemon juice in a second bowl. Mix bread crumbs and cheeses in a third bowl. Season chicken with salt and pepper.

Dredge chicken in flour, then egg mixture, and then bread crumb mixture. Place on a sheet pan.

Place breaded chicken in air fryer basket. Spray with nonstick spray. Cook at 400°F for 8-10 minutes, flipping halfway and spraying tops again. Repeat with remaining chicken.