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For the bechamel: Melt the butter in a non-stick pan over medium heat.
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Then whisk in flour.
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Continue whisking and cooking for 2-3 minutes.
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The flour mixture will be thick.
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Slowly whisk in milk, about 1/2 cup at a time.
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For the first cup, let the flour mixture absorb the milk before adding in more milk.
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It will first look like a paste, and then slowly thin out.
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Continue to cook over medium heat until the sauce thickens.
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This will take about 5-8 minutes.
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Once thick, whisk in the salt, nutmeg and Parmesan.
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Continue whisking until the cheese melts.
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Remove from the heat and cool slightly while you get your other sandwich ingredients ready.
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For assembling the sandwich: Line a baking sheet with parchment paper and preheat your oven to 325 F. Cut the brioche into 8 thick slices.
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Lay out 4 slices and top with the sliced or shredded Gruyere.
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Next, top the cheese with the ham slices and a spoonful of bechamel.
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Top with the other brioche slices.
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Gently spread the top side of the brioche with the softened butter.
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Preheat a large non-stick pan over medium low to medium heat.
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Place the brioche sandwiches into the pan with the buttered side down.
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While the brioche is browning, butter the other side of the sandwich.
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Cook for about 2 minutes, or when the bottom is golden.
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Once golden, flip the sandwich and brown the other side.
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Once the second side has browned, remove from the pan and place on the prepared baking sheet.
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Top each sandwich with 2 tablespoons of bechamel and 1 tablespoon of Parmesan cheese.
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Place in the oven to melt the cheese, about 5 minutes.
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Remove from the oven and set aside.
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Now make your fried egg.
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Today, I chose to do an over easy, but feel free to make a traditional sunny side up.
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Melt about 1 tablespoon of butter in a medium non-stick pan.
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Once the butter has melted, gently crack an egg into the pan.
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It should gently sizzle.
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Season with salt and pepper.
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Cook for about 90 seconds and then flip it over to cook the other side (about 90 more seconds).
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Cooking time will vary depending on how runny you want your yolk!
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Remove cooked egg(s) to a plate and finish cooking the rest.
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Top each sandwich with a fried egg and serve immediately!
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Slightly adapted from macrinabakery.com.
