The Croque Madame is a ham and cheesesandwich, dipped in egg and then broiled, topped with a fried eggà cheval, and topped with a Mornay or Béchamel sauce, served (hot) in the bistros of Paris.
The origin of the name is shrouded in mystery. Presumably it is derived from the croque Monsieur, or vice-versa. Croque comes from the verb croquer which means to crunch or to munch.
Ham is also a great match, which immediately brought to mind croque monsieur: a French-style ham and cheese (usually Gruyere) sandwich which is grilled, broiled or pan-fried in butter until the bread is golden brown and the cheese melted.
According to Alan Davidson's The Oxford Companion to Food, "croque monsieur literally means 'crunch-sir' or 'munch-sir' (croquer being the French verb 'to crunch' or 'to munch')." Its spouse -- the croquemadame -- is traditionally topped with a fried egg.
A classic croque monsieur is basically a ham and cheese sandwich made with crustless, buttered bread and broiled or pan-fried in butter.