Pound the chicken: Butterfly each chicken breast and place between sheets of plastic wrap. Pound to ¼-inch thickness. Season both sides with salt and pepper.
Stuff and roll: Layer 2 slices of ham and 2 slices of Swiss on each breast, leaving a border. Roll tightly from the narrow end and secure with toothpicks. Wrap in plastic wrap and refrigerate 30 minutes.
Dredge: Set up three shallow bowls with flour, beaten eggs, and panko. Coat each roll in flour, then egg, then panko, pressing so it sticks.
Preheat: Preheat the air fryer to 375°F for 3 to 5 minutes. Spray the basket with cooking spray.
Air fry: Place rolls seam side down in a single layer. Spray the tops with oil. Air fry at 375°F for 18 to 20 minutes, flipping halfway, until golden and the internal temperature reaches 165°F.
Make the sauce: Melt butter in a small saucepan over medium heat. Whisk in flour and cook 1 minute. Gradually whisk in milk and simmer until thickened, 3 to 4 minutes. Stir in Dijon and Parmesan. Season with salt and pepper.
Rest and serve: Rest the chicken 5 minutes, remove toothpicks, slice, and serve with the sauce.