A croque-monsieur served with a fried egg or poached egg on top is known as a croque-madame. The noted French chef Jacques Pepin also makes a version using chicken instead of ham, which he demonstrated in the “Our Favorite Sandwiches” episode on the PBS series (and its coordinating cookbook of the same title) Julia and Jacques Cooking at Home in which he worked with Julia Child. Many dictionaries attribute the name to the egg resembling an old fashioned woman’s hat. According to the Petit Robert dictionary, the name dates to around 1960.

I made one of each and added mustard for hubby on the ham.

CROQUE MADAME
PER PERSON
2 slices white or whole-wheat sourdough bread
2 ounces Gruyère cheese, sliced into 24-32 1/16-inch-thick slices
3 ounces smoked ham, such as Black Forest, sliced into thick slices
1 JUMBO egg
1 + 1 tablespoon unsalted butter
sea salt, to taste
freshly cracked black pepper, to taste

• Melt butter in griddle.
• Place half of the slices of bread in butter and cover them with the cheese slices, making sure they don’t extend past the edges of the bread.
• Place 2 slices of the ham in an even layer over the cheese, and place the top slice of bread over the ham.
• Grill the sandwiches, turning once.
• Place on serving plates.
• Crack eggs into separate bowls and check that the yolks aren’t broken.
• In a non-stick skillet, melt half of the butter over medium-high heat, until it starts to bubble.
• Add eggs into pan, being careful not to break the yolks or crowd the eggs.
• Add a teaspoon per egg of water to pan, sprinkle the eggs with salt and cover the pan with lid.
• Cook the eggs for about 3 minutes for a soft-cooked egg and 5-6 minutes if you like your eggs firm.
• Center a fried egg over each of the grilled sandwiches and sprinkle with pepper.