Pound chicken breasts to an even ¼-inch thickness between plastic wrap using a meat mallet. Season lightly with salt and pepper.
Lay one slice of ham on each pounded breast, then top with a slice of Swiss cheese, leaving a ½-inch border on all edges.
Roll each breast tightly from the shorter end, like a jelly roll. Secure with 3–4 toothpicks along the seam and refrigerate for 15 minutes.
Set up three shallow bowls: flour with salt and pepper; beaten eggs; and panko mixed with garlic powder, onion powder, smoked paprika, salt, and pepper.
Dredge each chilled roll in flour, dip in egg, then coat thoroughly with seasoned panko, pressing firmly. Spray all sides lightly with cooking oil.
Preheat air fryer to 375°F. Place rolls seam-side down in a single layer and air fry for 20–25 minutes, flipping halfway, until internal temperature reaches 165°F.
Melt butter in a small saucepan, whisk in flour and cook 1 minute. Gradually whisk in milk until smooth and thickened. Remove from heat and stir in Dijon mustard, garlic powder, salt, and pepper.
Let chicken rest 3–5 minutes, remove all toothpicks, slice diagonally, and serve with warm Dijon cream sauce. Garnish with fresh chives or parsley.