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1
Preheat the broiler.
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2
Make the bechamel sauce: In a small saucepan, melt 1 ounce butter over medium heat.
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3
Whisk in the flour and cook, stirring constantly, until you smell the flour and butter cooking, 2 to 3 minutes.
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4
Do not allow it to brown.
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5
Add the milk and bay leaf and cook, stirring from time to time, until the mixture thickens like a soup, 10 to 12 minutes.
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6
Remove the bay leaf and stir in the Parmesan.
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7
Transfer to a bowl to cool.
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8
Arrange 4 slices of the bread on a flat surface.
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9
Top each with 2 to 3 slices of ham.
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10
Mix together half of the Gruyere cheese and the bechamel sauce.
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11
Taste for seasoning.
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12
Spread a little of the sauce on top of the ham and top each with another slice of bread.
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13
Heat a large cast iron skillet and, when hot, add half of the remaining butter.
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14
Add 2 of the sandwiches and brown on one side, 2 minutes.
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15
Turn on the other side and brown 2 more minutes.
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16
Transfer to a baking sheet and repeat with the remaining butter and sandwiches.
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17
Spread the remaining bechamel on top of the sandwiches and top with the other half of the Gruyere cheese.
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18
Wipe any crumbs from the cast iron skillet and spray with nonstick spray.
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19
Crack 4 eggs into the skillet, leaving a little room between each.
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20
(Alternatively, fry 2 at a time).
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21
While the eggs are frying, place the sandwiches under the broiler and broil until the top becomes golden brown, 1 to 2 minutes.
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22
Season the eggs with salt and top each sandwich with a fried egg.
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23
Serve immediately.
