Preheat the oven to 375°F (190°C).
Toast the slices of bread until golden brown.
Spread Dijon mustard on one side of each slice of bread.
Layer ham on one slice of bread.
Add a slice of cheese on top of the ham.
Place the second slice of bread on top, mustard side down.
In a skillet, melt butter over medium heat and grill the sandwich until golden brown.
Flip the sandwich and grill the other side until cheese melts.
Transfer the sandwich to a baking dish and top with béchamel sauce.
Bake in the oven for about 10 minutes, until bubbly.
Fry an egg in the skillet until the white is set and yolk remains runny.
Top the baked sandwich with the fried egg and serve immediately.