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1
Preheat a two-sided electric griddle on medium-high to high for about 20 minutes, or preheat the oven to 300 degrees and preheat a cast-iron skillet on top of the stove for about 5 minutes.
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2
Prepare a bechamel sauce: in a small saucepan over medium heat, melt 1 tablespoon butter.
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3
When bubbles have subsided, add flour and whisk vigorously for 1 minute.
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4
Slowly whisk in milk until smooth.
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5
Bring to a boil, and cook until thick.
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6
Remove from heat, and season to taste with salt and nutmeg.
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7
Spread two slices of bread generously with sauce.
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8
Lay two slices of ham on top of each, and top each with a slice of cheese; ham and cheese should slightly overlap edges of bread.
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9
Top each with a slice of bread.
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10
In a small saucepan, melt remaining the 4 tablespoons butter.
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11
Brush the sandwiches on both sides with butter, making sure that the edges are well covered.
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12
If you're using a griddle, place cheese side down, close the griddle and cook until the bread is toasted dark and cheese is leaking out and bubbling.
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13
If you're using a skillet, place sandwiches cheese side down and cook on stove top until well browned, then turn and brown again.
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14
Transfer skillet to oven, and bake until heated through and cheese is bubbling.
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15
Meanwhile, fry the eggs in a skillet with little butter.
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16
Slide one fried egg onto each sandwich.
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17
Serve hot.
