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1
Melt the butter in a saucepan over medium heat. Add the flour all at once and stir constantly for one minute to cook the roux — it should smell faintly biscuity but carry no colour.
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2
Remove the pan from the heat and add the cold milk in a steady stream, whisking without pause. Return to medium heat and whisk constantly until the sauce thickens and coats the back of a spoon cleanly. Season with salt, white pepper, and nutmeg. Remove from the heat and stir in the 60 g of grated Gruyère until fully melted. The finished Mornay must be fluid enough to spread easily — a sauce that mounds or drags is too thick and will make the sandwich stodgy.
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3
Heat the oven to 220 °C. Lay the eight bread slices on a clean surface. Spread a thin, even layer of Mornay across one side of four slices (the bottom slices). Lay one slice of jambon de Paris flat on each. Set the four remaining slices on top to close the sandwiches. This is where the dish lives or dies: spread Mornay generously over the entire exterior — top face and all four sides — of each assembled sandwich, then press an even layer of grated Gruyère over the sauced surfaces. No bare patch of bread should remain exposed.
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4
Transfer the sandwiches to a flat baking sheet. Bake at 220 °C for 8 to 10 minutes, or finish under a salamander, until the cheese is deeply golden, bubbling across the entire surface, and lightly charred in irregular spots. Pale, unevenly melted cheese signals insufficient heat or time.
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5
For the croque-madame: while the croques bake, fry the eggs in butter over medium heat, basting the whites until set and crisp at the edge while the yolk remains fully runny. Place one egg on each finished croque immediately before serving — the broken yolk acts as a secondary sauce at table.