Croque Madame

Method

1. 1Make the béchamel first. Melt 20 g of butter in a small saucepan over medium-low heat. Add the flour all at once and whisk constantly for 2 minutes — the roux should look like a smooth, slightly golden paste and smell faintly of cooked flour (this step cooks out the raw flour flavor). Remove from the heat and add the milk in one go, whisking hard to prevent lumps. Return to medium heat and whisk constantly until the sauce thickens — about 3 to 4 minutes. It should coat the back of a spoon thickly. Season with a pinch of nutmeg, salt, and white pepper. Cover with cling film pressed directly onto the surface to prevent a skin forming. Set aside.
2. 2Toast the bread in butter. Preheat the grill (broiler) on high. Meanwhile, melt 30 g of butter in a large non-stick frying pan over medium heat. Working in batches if needed, toast both sides of all 4 bread slices in the butter until golden and slightly crisp — about 2 minutes per side. You are building the structural integrity of the sandwich here: untoasted bread goes soggy under the béchamel. Transfer the toasted bread to a baking sheet lined with foil.
3. 3Assemble the sandwiches. Spread a thin layer of Dijon mustard on one side of each slice of bread. Spread 1 to 2 tablespoons of béchamel over the mustard on 2 of the slices. Distribute the ham evenly over the béchamel, then scatter a third of the grated Gruyère over the ham. Place the remaining 2 slices on top, mustard-side down. Now spread a thick, generous layer of béchamel over the top of each sandwich — go to the edges. Scatter the remaining Gruyère over the béchamel top. Do not be stingy: the thick béchamel-cheese topping is the point.
4. 4Grill until bubbling and golden. Slide the baking sheet under the hot grill and cook for 3 to 5 minutes, watching constantly, until the Gruyère on top is melted, bubbling, and developing dark golden spots. The béchamel should puff slightly. Remove from the grill. Meanwhile, fry the eggs: melt the 10 g of butter for the eggs in the frying pan over medium-low heat, crack in the eggs, and cook until the whites are fully set but the yolks are still completely runny — baste with butter from the pan using a spoon if needed to set the whites without overcooking the yolks.
5. 5Place the eggs on top and serve immediately. Slide one fried egg on top of each grilled sandwich. The yolk must be runny — if it has set, the point of the dish is lost. Serve immediately on warmed plates with nothing else required, or alongside a lightly dressed green salad. This is a knife-and-fork sandwich. Cut into it first at the yolk: the yolk runs into the béchamel and becomes an additional sauce. That is the moment the whole thing comes together.