Make bechamel sauce by melting one tablespoon butter in a medium saucepan over medium heat. Whisk in flour until well combined. Gradually whisk in milk and cook,whisking,until mixture comes to the boil and thickens.
Simmer sauce for about 15 minutes,stirring every few minutes,then remove from heat. Season with salt and pepper.
Using remaining butter,butter each slice of bread. Take four slices,spread the unbuttered sides with mustard and top with salmon. Scatter dill over salmon. Top each with ¼ cup cheese. Close sandwich with remaining bread,buttered side up.
Preheat grill. Heat a large non-stick frypan over medium heat. Add sandwiches;cook,turning once,until golden brown,about two minutes each side.
Spread two tablespoons of bechamel sauce on top of each sandwich;sprinkle each with ¼ cup remaining cheese. Grill until cheese is bubbling and golden brown.
Meanwhile,heat a large non-stick frypan over medium heat. Crack eggs into pan. 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.