Step 1

Preheat the broiler.

Step 2

Make the bechamel sauce: In a small saucepan, melt 1 ounce butter over medium heat. Whisk in the flour and cook, stirring constantly, until you smell the flour and butter cooking, 2 to 3 minutes. Do not allow it to brown. Add the milk and bay leaf and cook, stirring from time to time, until the mixture thickens like a soup, 10 to 12 minutes. Remove the bay leaf and stir in the Parmesan. Transfer to a bowl to cool.

Step 3

Arrange 4 slices of the bread on a flat surface. Top each with 2 to 3 slices of ham. Mix together half of the Gruyere cheese and the bechamel sauce. Taste for seasoning. Spread a little of the sauce on top of the ham and top each with another slice of bread.

Step 4

Heat a large cast iron skillet and, when hot, add half of the remaining butter. Add 2 of the sandwiches and brown on one side, 2 minutes. Turn on the other side and brown 2 more minutes. Transfer to a baking sheet and repeat with the remaining butter and sandwiches. Spread the remaining bechamel on top of the sandwiches and top with the other half of the Gruyere cheese.

Step 5

Wipe any crumbs from the cast iron skillet and spray with nonstick spray. Crack 4 eggs into the skillet, leaving a little room between each. (Alternatively, fry 2 at a time). While the eggs are frying, place the sandwiches under the broiler and broil until the top becomes golden brown, 1 to 2 minutes. Season the eggs with salt and top each sandwich with a fried egg. Serve immediately.