I feel completely certain now that the croque madame is not a chicken sandwich.
The only Food Network show that I DVR is "Alex's Day Off." (I'm pausing for applause. That there's only one DVR commitment to this channel is something of an achievement for me.)
This morning, I found that she does the croque with a fried egg, too! Her sandwich is, of course, a touch fancier than my little open-faced eggy sammy, but not by much. She basically makes a cheesy bechamel to put inside two slices of bread, along with sliced ham. And on top, along with more bechamel and gruyere, she likes a perfectly fried egg.
There's no sign of a couple ingredients I tossed in, but I do like my Dijon mustard and lemon on this sandwich, so maybe I'll incorporate those into my bechamel instead of bay leaves like Alex does... Hm, actually, I wonder if lemon juice would curdle the bechamel or something.
One thing for certain: she also makes a champagne cocktail called a French 75 (a raw sugar cube, splash of gin, orange juice, lemon juice and champagne). I want one.