• 1 and 1/2 Tablespoons unsalted butter
• 1 and 1/2 Tablespoons all-purpose flour
• 1 cup whole milk
• 8 ounces Gruyère cheese, coarsely grated
• 2 ounces Parmesan cheese, finely grated
• Salt and freshly ground black pepper, to taste
• Pinch freshly grated nutmeg
• 4 English muffins, cut in half and toasted
• 4 Tablespoons Dijon mustard
• 8 slices ham
• 2 Tablespoons olive oil
• 4 large eggs
① In a medium saucepan over medium heat, melt the butter until it's lightly brown and smelling delicious. Add flour and cook, whisking constantly, until smooth, about 30 seconds. Whisk in the milk and bring the mixture to a boil, then reduce the heat to medium-low and simmer until slightly thickened, about 5 to 6 minutes. Add 1/4 cup of the grated Gruyère and all of the Parmesan, and whisk until smooth. Season with salt, pepper, and nutmeg.
② Preheat oven to broil setting. Place English muffins - cut side up - on a large baking sheet. Spread 1/2 a tablespoon of mustard onto each piece. Top 4 of the halves with 2 slices ham and then sprinkle with the remaining Gruyère, aiming to pile the cheese in the center of the muffin so it doesn't fall onto the baking sheet. Place the baking sheet under the broiler until the cheese begins to melt, about 1–2 minutes.
③ Top with remaining muffin slices, then pour a generous amount of béchamel on top of each muffin and broil until the cheese sauce is bubbling and evenly browned, about 4 minutes.
④ In the meantime, cook your eggs! Heat olive oil in a large nonstick skillet over medium heat. Add eggs, season with salt and pepper, and cook until whites are firm but yolks are still runny; about 3 minutes. Place an egg on top of each sandwich, and serve right away!
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