For the béchamel, melt the butter, cook the flour in it for 1 minute, stirring, then gradually pour in the cold milk and cook for 3-4 minutes into a thick sauce. Season with salt, white pepper and nutmeg.
Preheat the oven to 200 °C (grill). Butter the outsides of the bread, spread the insides of two slices with béchamel, add the ham and a quarter of the cheese, and cover with the other two slices.
Toast in a frying pan for 2 minutes per side until golden, put on a baking tray, spread the tops with the rest of the béchamel and sprinkle with the rest of the cheese.
Grill at the top of the oven for 5-7 minutes until the cheese bubbles and browns in patches.
Meanwhile, fry two eggs in the butter in a frying pan over medium-low heat for 3-4 minutes until the whites are set but the yolks runny. Salt lightly.
Slide the fried eggs on top of the sandwiches (this is what makes it a 'madame') and serve at once.
A Receptnaptár original recipe. Calories and protein are estimated.