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1
For the Mornay sauce: In a small saucepan over medium-low heat, make a roux by melting half of the butter and stirring in the flour.
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2
Cook for about 2 minutes and then slowly add the milk, whisking constantly for another 2 minutes.
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3
Reduce heat to low and simmer for about 5 minutes, whisking occasionally, until mixture thickens.
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4
Stir half of the cheese into the sauce then stir in the Dijon mustard, and salt, pepper and nutmeg to taste.
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5
Set aside.
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6
Assemble the sandwiches by spreading 1-2 tablespoons of the Mornay sauce on one side of each piece of bread, layer with the ham and remaining cheese and place together to form the sandwich.
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7
Spread remaining butter on the outside of the bread slices and place buttered side down in a skillet over medium-low heat.
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8
Cook until both sides are golden brown and cheese is melted inside.
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9
Transfer sandwiches to a baking sheet.
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10
Set skillet aside to cook the eggs.
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11
Preheat broiler.
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12
Divide the remaining Mornay sauce over the top of each sandwich and broil until the tops are browned and bubbling.
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13
While the sandwiches are broiling, lightly fry the eggs sunny-side up over medium heat in the skillet.
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14
Remove the sandwiches from the broiler and top each one with an egg.
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15
Use a fork and knife to devour the Croque-Madame and make another one because its so good!
