Prep

1. Butterfly slice the boneless skinless chicken breast horizontally, so that you have 4 thin chicken fillets.  Place each fillet between two sheets of plastic wrap and pound with a meat mallet until chicken is evenly pressed to 1/4 inch thick.
2. Season inside of chicken fillet with a light amount of kosher salt and black pepper. Spread yellow mustard overtop with a gloved finger or back of a spoon.  Break up 2 slices of swiss cheese per breast, and arrange so it sits in a flat layer edge to edge.  Next, do the same with a single slice of ham. 
3. Roll the chicken breast tightly vertically from the more narrow edge toward the top.  Place onto a large piece of plastic cling wrap. Roll the chicken in the wrap, leaving excess wrap on each side.  Twist the edges of the wrap in opposite directions, slowly pressing the chicken into a tight cylinder.  Tuck the twisted edged under the chicken and place it on a pan. Repeat with remaining breasts and place wrapped chicken in fridge for at least 30 minutes, up to overnight. 

Cook

1. Preheat air fryer to 400°.  Prepare two bowls for breading.  In the first bowl beat egg with cold water to form an egg wash.  In the second bowl, combine italian breadcrumbs and grated parmesan cheese.
2. To bread chicken, remove from wrap and roll first in the egg wash, covering all surfaces.  Lift chicken up and let excess egg drip back into the bowl.  Transfer the chicken to the breadcrumb bowl.  Roll it gently, then use a spoon to cover the top of the chicken with the breadcrumbs and gently press to help it to adhere to the chicken.  Stand the chicken on the end and cover each side.  Place on a pan and repeat with remaining chicken.
3. Place preheated chicken in the air fryer basket, making sure to leave space between them to avoid overcrowding.  Cook for 6 minutes, then open basket and flip chicken over.  Cook for 6 minutes more, until breadcrumbs have browned nicely and chicken is cooked through. Remove chicken from a basket and let cool for 10 minutes before serving.