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1
Heat the oven to 475 degrees F and arrange a rack in the middle.
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2
Place 1/3 cup of the cheese in a small bowl and set aside.
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3
Place the bread in a single layer on a baking sheet.
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4
Melt 5 tablespoons of the butter and brush it on both sides of the bread, using all of it.
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5
Bake until the bread is toasted and golden brown, about 10 minutes.
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6
Set the baking sheet and bread aside.
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7
Meanwhile, make the bechamel: Melt the remaining tablespoon of butter in a small saucepan over medium heat until foaming.
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8
Whisk in the flour and cook, whisking occasionally, until its no longer raw-tasting, about 30 seconds.
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9
Slowly whisk in the milk and add the measured salt and nutmeg.
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10
Cook, whisking constantly, until the mixture thickens, about 3 minutes.
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11
Remove the pan from the heat and set it aside.
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12
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.
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13
Spread a thin layer of Dijon on the remaining 4 slices of bread and place them mustard-side down on top of the cheese.
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14
Press gently to compact the sandwiches.
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15
Cover the baking sheet tightly with aluminum foil and bake until the cheese is just starting to melt, about 5 minutes.
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16
Remove the baking sheet from the oven and discard the foil.
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17
Whisk the reserved bechamel sauce until smooth, then spread it on the tops of the sandwiches, using all of it.
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18
Sprinkle the reserved 1/3 cup of cheese over the bechamel.
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19
Bake the sandwiches uncovered until the cheese on top is melted and bubbling, about 10 minutes.
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20
Meanwhile, crack 8 of the quail eggs into a small bowl.
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21
When the sandwiches are ready, remove them from the oven and set aside.
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22
Heat the vegetable oil in a large nonstick frying pan over medium heat until shimmering.
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23
Pour the 8 cracked eggs into the pan, spacing them evenly (if some of the whites stick together, separate them with a spatula).
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24
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.
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25
Remove the fried eggs to a large plate.
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26
Repeat with the remaining 8 eggs.
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27
Trim the crusts from each reserved sandwich, then cut each sandwich into 4 smaller squares.
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28
Place the sandwiches on a serving platter, top each with a fried quail egg, and serve immediately.
