The Croque-Madame is the Croque-Monsieur with a fried egg on top, and the entire identity of the sandwich hangs on the yolk. The base is unchanged: pain de mie, jambon de Paris, Gruyère or Comté, a coat of béchamel, the top blistered under the broiler. What turns it into a different sandwich is the egg, fried separately to set the white while keeping the yolk loose, then crowned on the bubbling top. The name borrows the visual logic of a hatted lady. When the sandwich arrives at the table the diner punctures the yolk with a fork and lets it run, and the loose yolk pools into the cheese, thins the béchamel, and pushes the whole assembly somewhere richer than the Monsieur ever goes. A barely-set yolk is the point. A yolk cooked hard is a different sandwich entirely, and a lesser one.
The structure rewards the egg in a way that surprises first-time eaters. The béchamel and the running yolk are doing a similar job in different registers: both are emulsified, both coat the bread and bind the layers, but the yolk brings sulfur, fat, and a faint sweetness the béchamel does not. The result is a sandwich that has to be eaten with a knife and fork, on a plate, with the broken yolk allowed to run wherever it wants. It is also a sandwich that does not travel. Within five minutes of leaving the broiler the egg sets, the béchamel cools, and the magic is over. The Madame is a sit-down lunch in a café with a zinc counter and a service window open to the kitchen, and nothing else.
The Croque-Madame is documented on Parisian café menus going back to the early twentieth century, the egg-on-top variant emerging alongside the Croque-Monsieur itself and earning its own name fast. From there it has spawned its own sub-branches: the Croque-Madame Parisien emphasizes the runny-yolk school, and various regional kitchens stack the egg on a Savoyard or Provençal base instead of the classique. The broader cluster of croque variants belongs to its own context, mapped at Croque-Monsieur. The Madame is the most recognizable branch.