Make the bechamel sauce: Melt 1 tablespoon butter in a wide medium saucepan over medium-high heat. Whisk in flour; cook, whisking, 30 seconds. Gradually whisk in milk; cook, whisking, until mixture comes to a boil and has thickened, about 2 minutes. Remove from heat. Season with salt and pepper.
Using remaining 4 tablespoons butter, butter 1 side of each piece of bread. Set aside 4 slices. Spread unbuttered sides of remaining 4 slices with mustard; top with salmon. Scatter dill over salmon. Top each with 1/4 cup cheese. Sandwich with remaining bread, buttered side up.
Preheat broiler. Heat a large nonstick skillet over medium-low heat. Add sandwiches; cook, turning once, until golden brown, about 2 minutes per side. Spread 2 tablespoons reserved bechamel sauce on top of each sandwich; sprinkle each with 1/4 cup remaining cheese. Broil until cheese is bubbling and golden brown.
Meanwhile, heat a large nonstick skillet over medium heat. Crack eggs into skillet. Cover, and cook until whites are set but yolks are still soft, about 2 minutes. Season with salt and pepper. Place 1 egg on top of each sandwich, and serve.