Steps
1. Preheat the oven to 180°C (350°F).
2. In a small saucepan, melt 1 tablespoon of butter over medium heat. Add the flour and whisk continuously for about 2 minutes until the mixture turns golden.
3. Slowly pour in the milk while whisking constantly to avoid lumps. Cook the béchamel sauce for 5-7 minutes until thickened. Season with salt and pepper.
4. In a separate skillet, melt the remaining butter over medium heat. Place one slice of bread in the skillet and top with a slice of ham and cheese. Cover with the second slice of bread.
5. Cook the sandwich for 2-3 minutes on each side until golden brown. Remove from the skillet.
6. In the same skillet, fry the eggs to your desired doneness.
7. Place the sandwich on a baking sheet and pour the béchamel sauce over the top. Place under the preheated oven for 5 minutes until the sauce is bubbly and golden.
8. Transfer the croque madame to a plate, top with the fried eggs, and season with salt and pepper.
9. Serve hot and enjoy!