In a medium saucepan, melt 3 tablespoons of butter. Stir in flour until completely blended, then cook over moderately hight heat, stirring constantly, until roux is golden brown, about 3 minutes. Gradually whisk in milk until smooth. Bring sauce to boil, whisking constantly. Reduce heat to moderately low and simmer, whisking often, until no floury taste remains, about 5 minutes. Remove saucepan from heat and let sauce cool, stirring, until very warm. Stir in chopped salmon, Gruyeres, chives, and nutmeg. Season with salt and pepper and let 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 remaining salmon mixture. Can cover sandwiches with plastic wrap and store in refrigerator overnight, if desired.
Preheat oven to 400°F In a large shallow bowl, beat eggs thoroughly. Stack 2 or 3 large rimmed baking sheets on one side of the stove.
In a large skillet, melt 1 tablespoons of butter over moderate heat. Dip 8 of the sandwiches in the beaten egg to coat completely, cook the egg-dipped sandwiches over moderately-high heat, turning once, until browned, about 1 minutes per side. Transfer sandwiches to one of the baking sheet, leaving a little space between each one. Repeat with the remaining sandwiches and beaten eggs, using 1 tbsp of butter per batch.
Reheat sandwiches in the oven for about 8 minutes, until piping hot. Cut sandwiches in half on the diagonal, skewer with small wooden forks and serve hot.
Sandwiches can be made earlier in the day and reheated.