This is how Wikipedia define Croque Madame - A croque-monsieur served with a fried egg or poached egg on top is known as a croque-madame.
A croque-monsieur is a hot ham and cheese (typically emmental or gruyere) grilled sandwich. It originated in France as a fast-food snack served in cafes and bars. More elaborate versions come coated in a Mornay or Bechamel sauce. The emergence of the croque-monsieur (and variations) is mirrored by growth in popular fast-foods in other countries.
The name is based on the verb croquer ("to crunch") and the word monsieur ("mister")—the reason behind the combination of the two words is unclear—and is colloquially shortened to croque. While the origins of the croque-monsieur are unknown, there are many speculations on how it was first originated. The croque-monsieur's first recorded appearance on a Parisian cafe menu was in 1910. Its earliest published use has been traced back to volume two of Proust's Remembrance of Things Past (Ala recherche du temps perdu) (1918)."
Ingredients for Bechamel Sauce
2 tbsp butter
1 tbsp plain flour
1 cup milk
2 tsp mustard
salt and pepper to taste
Melt butter in a sauce pot. Stir in flour and cook for 1 minute. Whisk in milk and bring to bubble, then reduce the heat. Add in mustard, salt and pepper. Stir till well combine and slightly thicken.
Ingredients for 2 sets of Open Sandwiches
2 slices of white bread
2 tbsp butter
4 slices of ham, shredded
2 slices of Gouda cheese
1. Apply butter on one side of the bread and set aside.
2. Heat a non-stick pan and fry the 2 eggs separately.
3. Remove eggs from pan and transfer bread with buttered side on non-stick pan.
4. Top with a slice of cheese, shredded ham, bechamel sauce and egg.
5. Cover and let sandwich heat through till cheese has melted.
6. Serve hot with more bechamel sauce if required.