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1
Make the bechamel sauce: Melt 1 tablespoon butter in a wide medium saucepan over medium-high heat.
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2
Whisk in flour; cook, whisking, 30 seconds.
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3
Gradually whisk in milk; cook, whisking, until mixture comes to a boil and has thickened, about 2 minutes.
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4
Remove from heat.
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5
Season with salt and pepper.
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6
Using remaining 4 tablespoons butter, butter 1 side of each piece of bread.
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7
Set aside 4 slices.
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8
Spread unbuttered sides of remaining 4 slices with mustard; top with salmon.
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9
Scatter dill over salmon.
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10
Top each with 1/4 cup cheese.
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11
Sandwich with remaining bread, buttered side up.
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12
Preheat broiler.
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13
Heat a large nonstick skillet over medium-low heat.
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14
Add sandwiches; cook, turning once, until golden brown, about 2 minutes per side.
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15
Spread 2 tablespoons reserved bechamel sauce on top of each sandwich; sprinkle each with 1/4 cup remaining cheese.
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16
Broil until cheese is bubbling and golden brown.
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17
Meanwhile, heat a large nonstick skillet over medium heat.
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18
Crack eggs into skillet.
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19
Cover, and cook until whites are set but yolks are still soft, about 2 minutes.
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20
Season with salt and pepper.
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21
Place 1 egg on top of each sandwich, and serve.
