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Place each chicken breast on a cutting board between 2 pieces of parchment paper. Being careful not to pound all the way through, use a mallet or a foil-wrapped brick to flatten each breast to a thickness of 1/4 inch.
Cut ham and Swiss cheese slices in half so you have 4 pieces of each. Stack 1 piece ham and 1 piece of cheese in the middle of each flattened chicken breast. Roll up like a jelly roll, tucking the ham and cheese inside the breast as you roll. Secure the ends and sides with toothpicks so the cheese won't run out. Transfer to the refrigerator for at least 15 minutes, or up to 4 hours.
Preheat an air fryer to 350 degrees F (175 degrees C), if recommended by the manufacturer.
Combine flour, salt, and pepper in a small bowl. Place beaten egg in second bowl. Place bread crumbs in a third bowl.
Dip each chicken bundle in flour mixture, then in egg, and finally in bread crumbs to coat. If you notice any spots that are just coated in flour, lightly pat with beaten egg and bread crumbs. Gently pat each bundle with melted butter and place in the air fryer basket.
Air-fry until crust is nicely browned and chicken is no longer pink in the center and the juices run clear, about 20 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).