Make the béchamel

• Melt a little butter in a saucepan and stir in the flour.
• Pour in the milk gradually while whisking until the sauce thickens.
• Season with salt and pepper and stir in part of the grated cheese.

Assemble the sandwiches

• Spread the remaining butter and the mustard on the bread slices.
• Layer the ham and some grated cheese between two slices per sandwich.

Bake the croques

• Place the sandwiches on a baking tray and coat the tops with béchamel.
• Scatter the rest of the cheese over the top.
• Bake at 200 °C until golden and bubbling.

Fry the eggs

• Fry the eggs in a hot pan until the white is set and the yolk still runny.

Serve

• Top each baked croque with a fried egg and serve immediately.