Preheat a 6-quart air fryer to 370°F for 3 minutes. Cut a horizontal pocket into each breast from the thick side, leaving the opposite edge attached.
Spread Dijon inside the pockets. Wrap Swiss sticks in ham, tuck into the pockets and close with wooden toothpicks.
Put flour in one shallow dish, beat egg with water in a second and mix breadcrumbs, salt and pepper in a third. Coat chicken in flour, egg and crumbs and dab all surfaces with oil.
Arrange the breasts with space between them. Air fry for 12 minutes, turn carefully and cook for 10 to 15 minutes more until the thickest chicken beside the pocket and the filling center reach 165°F.
Rest for 5 minutes, remove every toothpick and slice across the pockets to show the ham and cheese.