Melt the butter in a pan, add the flour and whisk. Gradually add the milk, mix well and bring to boiling point.
Season with salt and add a little nutmeg.
Set aside. Pre-heat the appliance. Spread the bechamel sauce and grated cheese and place one slice of ham onto each slice of bread.
Place the remaining slices of bread on top.
Toast them in the appliance for 5 minutes.
Meanwhile, fry the eggs in a frying pan with a little oil for 5 minutes.
Place one fried egg on top of each croque and serve immediately.