Preheat a 6-quart air fryer to 370°F for 3 minutes. Cut horizontal pockets from thick side of each breast, keeping opposite edges attached.
Spread Dijon inside pockets. Wrap Swiss sticks in ham; tuck into pockets and secure with wooden toothpicks.
Put flour in a shallow dish, egg beaten with water in another, and breadcrumbs mixed with salt and pepper in a third. Coat chicken in flour, egg, then crumbs; dab all surfaces with oil.
Space breasts in basket. Air fry 12 minutes; carefully turn and cook 10 to 15 minutes more until thickest chicken beside pocket and filling center reach 165°F.
Rest 5 minutes; remove every toothpick. Slice across pockets to reveal ham and cheese.