Croque madames, toasted French ham and cheese sandwiches topped with béchamel sauce and fried eggs, work just as well as dainty appetizers. Instead of frying each sandwich individually the traditional way, use the oven to toast the bread, heat up the ham and béchamel, and melt the cheese for all of the sandwiches at one time. Cut them into bite-sized portions and top with tiny fried quail eggs for an irresistible appetizer for cocktail parties.
- Heat the oven to 475°F and arrange a rack in the middle. Place 1/3 cup of the cheese in a small bowl and set aside.
- Place the bread in a single layer on a baking sheet. Melt 5 tablespoons of the butter and brush it on both sides of the bread, using all of it. Bake until the bread is toasted and golden brown, about 10 minutes. Set the baking sheet and bread aside.
- Meanwhile, make the béchamel: Melt the remaining tablespoon of butter in a small saucepan over medium heat until foaming. Whisk in the flour and cook, whisking occasionally, until no longer raw-tasting, about 30 seconds. Slowly whisk in the milk and add the measured salt and nutmeg. Cook, whisking constantly, until the mixture thickens, about 3 minutes.
- Remove the pan from the heat and set aside.
- Evenly divide the ham over 4 of the toasted bread slices, then evenly divide the remaining 1 1/3 cups of cheese over the ham.
- Spread a thin layer of Dijon on the remaining 4 slices of bread and place them mustard-side down on top of the cheese. Press gently to compact the sandwiches. Cover the baking sheet tightly with aluminum foil and bake until the cheese is just starting to melt, about 5 minutes.
- Remove the baking sheet from the oven and discard the foil. Whisk the reserved béchamel sauce until smooth, then spread it on the tops of the sandwiches, using all of it. Sprinkle the reserved 1/3 cup of cheese over the béchamel. Bake the sandwiches uncovered until the cheese on top is melted and bubbling, about 10 minutes. Meanwhile, crack 8 of the quail eggs into a small bowl.
- When the sandwiches are ready, remove them from the oven and set aside. Heat the vegetable oil in a large nonstick frying pan over medium heat until shimmering. Pour the 8 cracked eggs into the pan, spacing them evenly (if some of the whites stick together, separate them with a spatula).
Season the eggs with salt and cook undisturbed until the whites are just set but the yolks are still runny, about 2 to 3 minutes. Remove the fried eggs to a large plate. Repeat with the remaining 8 eggs.
- Trim the crusts from each reserved sandwich, then cut each sandwich into 4 smaller squares.
- Place the sandwiches on a serving platter, top each with a fried quail egg, and serve immediately.