Croque madame is a classic French sandwich filled with ham and bathed in a warm, cheesy bechamel sauce. Add a fried egg and you've got croque madame. I've made this sandwich endlessly, but never for a group because you can only prepare them one at a time. Then I created this casserole--aka "hot dish," the Midwestern term--of white toast layered with bechamel, ham, and shredded Gruyere and topped with a few runny eggs.
Preheat the oven to 375 degrees F. Generously grease a 9 by 13-inch glass or ceramic baking dish with butter.
Spread the bread on a large baking sheet and bake for about 10 minutes, until dry and very lightly toasted.
Meanwhile, in a medium saucepan, melt the butter over medium-high heat. Add the flour and cook, whisking continuously, until a paste forms and it starts to bubble, about 2 minutes. Gradually whisk in the milk until smooth. Bring the sauce to a boil and then simmer over medium-low heat until thickened and no floury taste remains, about 7 minutes. Whisk in the mustard and season the sauce with salt and pepper.
Spread 1/2 cup of the sauce in the prepared baking dish. Arrange half the toast in the baking dish and spread half the remaining sauce on top. Scatter half the ham and cheese on the coated bread. Repeat the layering one more time with the remaining bread, sauce, ham and cheese.
Bake the hot dish for 15 minutes, until hot and the cheese has melted. One at a time, crack the eggs into a small bowl, then slide them onto the hot dish so that the whites spread a little and the yolks sit in little divots; use the back of a spoon to make divots if the arrangement of your bread doesn't naturally make them. Bake the hot dish for 12 to 15 minutes more, until the egg whites are just set but the yolks are still runny. Serve hot, sprinkled with more salt and pepper.
Do it ahead: You can make the creamy sauce the day before and store it in an airtight container in the refrigerator. Reheat it gently in the microwave before assembling the casserole.