Put a fried egg on a croque monsieur and you have a croque madame, the version that turns up on Japanese cafe menus alongside its eggless sibling. The base is the same: ham and cheese between bread, blanketed in bechamel and browned under heat until the top blisters. The egg on top is the whole distinction, and it is not a garnish. Its runny yolk becomes a second sauce, threading through the bechamel and the melted cheese so the dish reads richer and looser than the monsieur. The fried white gives a crisp lacy edge; the yolk gives a golden flood. Together with the salty ham and the toasted bread underneath, the egg is what tips this from a hot sandwich into something closer to a plated brunch.
The craft is the croque monsieur craft plus one more thing done well: the egg. The bread is a thick-cut shokupan or a sturdy white slice, toasted enough to stay firm under all that moisture. Ham and a good melting cheese go between, bechamel over the top, a little more cheese, then heat until the surface bubbles and browns. The egg is fried separately and slid on at the end so the white is set with a crisp frill while the yolk stays liquid; an egg cooked hard on top loses the entire reason it is there. The bind has to carry extra weight now, so the bechamel must be thick enough not to be thinned into a puddle by the yolk, and the bread toasted enough not to drown. A good one cuts to reveal yolk running down through sauce and cheese onto a base that still has structure. A sloppy one is a hard overcooked egg sitting on a soggy raft, or a thin bechamel and a broken yolk that together turn the whole plate to wet beige.
The variations mostly stack onto the monsieur's variations, with the egg riding along. Cafes add sauteed mushrooms or tomato beneath the bechamel, swap in a sharper cheese, or finish with chives and pepper for an adult brunch register. Some serve it open-faced so the egg sits proud and photogenic; some keep it closed for a tidier hand-held thing. The eggless parent, the plain croque monsieur with its ham, cheese, and bechamel and none of this yolk business, is the other half of this story and deserves its own article rather than being crowded in here.