4 cups (950 ml) grated Comte cheese (or other milder white melting cheese such as Fontina)
4 large eggs
Dill, to garnish
Melt butter in a wide medium saucepan over medium-high heat. Whisk in flour; cook, whisking, 30 seconds. Gradually whisk in the milk and cook, continuing to whisk, until mixture comes to a boil and has thickened, about two minutes. Remove from heat. Season with nutmeg, salt, and pepper.
Butter one side of each piece of bread. Spread unbuttered sides of all 8 slices with mustard- top half with four slices salmon. Sprinkle 1/4 of the minced capers over the salmon, season with pepper. Top each with 1/2 cup (120 ml) cheese. Sandwich with remaining bread, buttered side up.
Preheat broiler. Heat a large non-stick skillet over medium-low heat. Add sandwiches; cook, turning once, until golden brown, about two minutes per side, then transfer to a baking sheet. Spread 1/4 of the reserved bechamel sauce on top of each sandwich; sprinkle each with 1/2 cup (120 ml) remaining cheese. Broil until cheese is bubbling and golden brown.
Meanwhile, heat a large non-stick skillet over medium heat. Crack eggs into skillet. Cover, and cook until whites are set but yolks are still soft, about two minutes. Season with salt and pepper. Place one egg on top of each sandwich, and serve. Garnish with sprigs of fresh dill.