Bring your knife and fork to the table for this classic French ham and cheese sandwich, because things are about to get messy. A croque madame is a croque monsieur with a fried egg plonked on top of it. Apparently it acquired its “madame” title because the sunny-side-up egg resembles a lady’s hat—although it doesn’t keep its form for long when I’m around to destroy it.
To make the Mornay sauce, in a medium heavy saucepan, melt the butter over medium-high heat. Stir in the flour and cook, stirring frequently, for about 2 minutes, or until the butter and flour are well-blended and beginning to smell toasted. Gradually stir in half of the milk and bring the mixture to a boil, stirring constantly to prevent lumps. Stir in the remaining milk and bring back to a boil, stirring constantly. Simmer, stirring, for 3 minutes, or until the sauce has thickened. Remove from the heat and stir in the Gruyère until melted. Stir in the Dijon mustard and nutmeg, and season with salt and pepper. Cover to keep the sauce warm.
Make-Ahead: The Mornay sauce can be made up to 2 days ahead, cooled, covered, and refrigerated. Rewarm in a heavy saucepan over medium heat, stirring, for about 3 minutes and adding more milk to thin the sauce to the desired consistency, if necessary.