How to Make It
In a medium saucepan, melt 3 tablespoons of the butter. Stir in the flour until completely blended, then cook over moderately high heat, stirring constantly, until the roux is golden brown, about 3 minutes. Gradually whisk in the milk until smooth. Bring the sauce to a boil, whisking constantly. Reduce the heat to moderately low and simmer, whisking often, until no floury taste remains, about 5 minutes. Remove the saucepan from the heat and let the sauce cool, stirring, until very warm. Stir in the smoked salmon, Gruyère, chives and nutmeg. Season with salt and pepper and let the sauce cool completely.
Spread half of the rye slices with a rounded tablespoon of the salmon mixture and close the sandwiches. Repeat with the pumpernickel slices and the remaining salmon mixture.
Preheat the oven to 400°. In a large, shallow bowl, beat the eggs thoroughly. Stack 2 or 3 large rimmed baking sheets on one side of the stove.
In a large skillet, melt 1 tablespoon of the butter over moderate heat. Dip 8 of the sandwiches in the beaten eggs to coat completely; cook the egg-dipped sandwiches over moderately high heat, turning once, until browned, about 1 minute per side. Transfer the sandwiches to one of the baking sheets, leaving a little space between each one. Repeat with the remaining sandwiches and beaten eggs, using 1 tablespoon of butter per batch.
Reheat the sandwiches for about 8 minutes, until piping hot. Cut the sandwiches in half on the diagonal, skewer with small wooden forks and serve hot.