Preheat the oven to 350°F (180°C). Grease a 6-cup muffin tin well.
Cut the crusts off the bread slices and gently flatten each slice.
Brush the bread with melted butter, then press each slice into a muffin cup.
Bake the bread cups for 5 minutes to help them hold their shape.
Melt 1 1/2 tablespoons butter in a small saucepan over medium heat.
Add the flour and whisk for 1 minute.
Slowly pour in the milk while whisking until smooth.
Cook until the sauce thickens, then stir in cheese, Dijon mustard, nutmeg, salt, and black pepper.
Add chopped ham into each bread cup.
Spoon a little cheese sauce over the ham.
Crack 1 egg into each cup.
Sprinkle extra grated cheese on top.
Bake for 15 to 20 minutes, or until the egg whites are set and the cheese is melted.
Garnish with parsley and a little nutmeg. Serve warm.