Preheat a 6-quart air fryer to 350°F for 3 minutes. Spread Dijon over the inner faces of 2 bread slices.
Layer half the Edam, the ham and remaining Edam over those slices, then close with the other slices. Spread mayonnaise thinly over the outside faces.
Secure the top bread slice of each sandwich with 1 wooden toothpick. Place one sandwich in the basket and cook for 4 minutes.
Remove the toothpick completely, turn the sandwich carefully and cook for 3 to 5 minutes more until the bread is golden, cheese is melted and the ham filling reaches 165°F. Repeat with the second sandwich.
Check that both toothpicks have been removed, rest sandwiches for 1 minute and cut in half before serving.