French recipes, in French, from France, omit any sauce inside or out, and give the sandwich just as Robert defined it. (cf. Le Journal des Femmes. http://www.linternaute.com/femmes/cuisine/recette/283731/2545434833/croque-madame.shtml . Valid as of June 2009.)
This hasn’t stopped American innovation on the other side of the pond, however.
Ronald E. Cohen, writing for the United Press in 1974, says a Croque Madame is an open-faced sandwich with a cheese sauce on top of ham:
“Croque Madame: For each sandwich, a long slice of French bread or half a hero roll. Dry the bread out in the oven, 350 degrees, for about two minutes. Remove the bread and place a moderately thick piece of boiled or baked ham on it. Spread a cheese topping over the ham. Mom can do this and leave it in the refrigerator. The ingredients for the cheese topping are one cup of grated Swiss, one egg, one tablespoon flour. One-third cup of beer. Don’t worry Mom, it’s precious little beer. It cannot be tasted in the sandwich, and French kids have been eating this for decades with no visible effects. But if it makes you nervous, milk is acceptable. Mix the topping thoroughly, sprinkling in a touch of fresh ground pepper and a pinch of Tabasco sauce. After spreading on the ham, brown it for 15-20 minutes in a 450-degree oven.” 
Another popular American variant is a Croque Madame defined as a Croque Monsieur, but with sliced chicken instead of ham: Sharon Tyler Herbst says in the Food Lover’s Companion that “in France, this is a croque monsieur (toasted ham and cheese sandwich) with the addition of a fried egg. In Britain and America, a croque madame simply substitutes sliced chicken for the ham, with no sign of an egg.” 
Yet another American variant combines the chicken *and* the egg: “A croque madame, sometimes made with chicken instead of ham, is served with a fried egg on top.” 
And finally, another American variant defines it as a cooked Croque Monsieur that is topped with a Mornay or Béchamel sauce (there are battles within this camp), broiled and then topped with a fried egg. Some within this camp also sneak some sauce inside the sandwich as well. 
The confusion, though, seems largely confined to America. The French still seem to know exactly what a Croque Madame is.
Croque-mademoiselle can mean just about anything, anywhere.
The name dates to around 1960 (according to the Petit Robert dictionary.)
Literature & Lore
The Jamaicans may have scored the prize for the liveliest variation on the left side of the pond:
“Lovely big fat Jamaica plantains are readily available in our good markets. And from the use of these valuable banana relatives, the inventive cook can make all sorts of lovely edibles. Here, for instance, is an unctuous hot hors d’oeuvre, based on the classic Croque Madame, which I consider an exciting addition to the Holiday menu.
CROQUE JAMAICA (Per serving)
2 tablespoons fork-mashed very ripe plantain
2 thinly-sliced firm white bread crusts trimmed off
1 thin slice Swiss or Gruyere cheese
1 thin slice cooked ham
Prepared mustard to taste
2 large well-beaten eggs
Pinch ground allspice (Jamaica pimento)
Hot deep oil or fat for deepfrying (heated to about 370 degrees)
Trim off crusts from 2 thinly sliced slices of firm white bread, if possible a day old. Spread rather liberally on one side only with softened butter, then divide fork-mashed very ripe Jamaica plantain, and spread on both this side and the other. Arrange the thin slice of cheese, and then that of the cooked ham (baked if possible), and spread over ham prepared mustard of your choice to taste. Top with other slice of bread, and press firmly into a good sandwich. Cut croque with a sharp knife into three lengths, then into crosswise halves, again pressing each piece firmly together. In a shallow bowl beat the eggs with the pinch of allspice, and in this mixture dredge the small sandwiches a couple at a time. In a sizeable kettle, heat enough oil or fat to afford at least two inches of depth to about 378 degrees Fahrenheit. Gently lower the tiny sandwiches into this mixture, two or three at a time, to fry until nicely browned on both sides, turning them carefully with slotted spoon to accomplish this. Remove with slotted spoon or spatula, drain well on absorbent paper towels, and serve while hot, as special snacks.” — “Gourmet Chef”. “Make Use of Indigenous Vegetables.” Kingston, Jamaica: The Gleaner. 4 December 1969. Page 22.
Croque Madame is called a “croque-cheval” in some parts of Normandy.
 Cohen, Ronald E. Chicago, Illinois: Daily Herald. 23 May 1974. Page 67.
 Herbst, Sharon Tyler. Food Lover’s Companion, Third Edition. Barron’s Educational Series, Inc. 2001.
 Fabrigant, Florence. C’est si bon: Croque monsieur, madame showcase ham and cheese a la Francaise. New York: Nation’s Restaurant News. 11 September 2000.
 Croque Madame recipe. Unknown author. Epicuirous March 2007.