This is a classic. Belonging to a time of early Café society, bistro-lined boulevards and mentions in Proustian works next to madeleines, the croque monsieur (the madame has an egg on it) is like the grilled cheese sandwich of France. Its name literally means, "Mr. Crunch--or in this case, "Mrs. Crunch," due to the blistered bechamel covering crusty, toasted bread. Typically served with Parisian ham, I adjusted mine a bit with prosciutto to make the most of the crunch factor. Once out of the oven and topped with an egg, it almost takes on the attitude of a savory French toast. This version is already decadent and delicious but if you want to really send this over the top, serve it with homemade French fries and extra bechamel for dipping. And then take a nap.
Preheat the oven to 400 degrees F. Line a rimmed baking sheet with aluminum foil.
Combine the 3 1/2 cups milk, parsley, peppercorns, chile flakes, nutmeg, bay leaf and a pinch of salt in a medium saucepan and bring to a simmer.
Melt 4 tablespoons of the butter in another medium saucepan over medium heat, then add the flour, whisking to combine. Cook, whisking, until the mixture is doughy and smells a bit nutty, about 1 minute. Add the hot milk through a fine-mesh sieve about a half cup at a time, whisking to incorporate with the flour and butter mixture. The resulting bechamel should be smooth and velvety. If it's chunky or too thick, whisk in more milk to thin it out. Remove the bechamel from the heat and stir in half of the cheese.
Spread the Dijon on the sourdough toasts and transfer to the prepared baking sheet. Top each toast with 3 slices of the prosciutto and the remaining cheese. Spoon about 1/4 cup bechamel onto each toast and bake for 5 minutes. With the toasts still in the oven, switch the oven to broil and bake until the tops are bubbling and lightly browned, about 3 minutes.
Meanwhile, melt 1/2 tablespoon of the remaining butter in a small nonstick skillet over medium heat and add 1/2 tablespoon of the olive oil. Crack 1 egg into a cup then carefully add to the hot skillet (the cup helps control the egg). Season the egg with salt and pepper then spoon the butter and oil in the skillet over the egg until cooked to your liking. Top 1 toast with the fried egg. Repeat with the remaining eggs and serve immediately.
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