Preheat and prep: Preheat your air fryer to 360°F (182°C). Lightly spray the basket to prevent sticking.
Make the béchamel: In a small saucepan, melt 2 tablespoons of butter over medium heat. Whisk in 1 tablespoon of flour and cook for 1 minute until foamy, not browned.
Slowly whisk in warm milk. Simmer and whisk until thick, 2–3 minutes. Season with salt, pepper, a pinch of nutmeg, and 1 teaspoon Dijon.
Remove from heat.
Assemble the base: Spread a thin layer of Dijon on one side of each bread slice. Place a slice in the air fryer-safe tray or on a small piece of parchment. Add a spoonful of béchamel, a generous handful of grated cheese, and the ham.
Top with the second slice of bread. Press gently.
Top with more béchamel: Spread a thin layer of béchamel over the top of the sandwich and sprinkle with more cheese. This helps create that golden, bubbly crust.
Air fry the sandwich: Cook at 360°F (182°C) for 6–8 minutes until the cheese is melted and the top is starting to brown.
If the top is browning too fast, tent loosely with a small piece of foil.
Cook the egg: While the sandwich finishes, heat a small nonstick pan with a touch of butter or oil. Fry the egg to your preference—sunny-side up with a runny yolk is classic. Season lightly with salt and pepper.
Alternatively, you can crack the egg into a lightly oiled, air fryer-safe ramekin and cook in the air fryer at 320°F (160°C) for 5–7 minutes until set but still runny.
Finish and serve: Remove the sandwich when the cheese is bubbling and edges are crisp. Top with the fried egg. Sprinkle with chives or parsley if you like.
Serve hot.
For extra toastiness: If you prefer, lightly butter the outside of the bread before assembling for a richer crust. Adjust time by 1–2 minutes if needed.