
Make the Béchamel:In a saucepan, melt butter over medium heat. Add flour and whisk for 1–2 minutes (don’t let it brown). Slowly pour in milk while whisking. Let simmer for 5–7 minutes, whisking constantly, until it thickens to a gravy-like consistency. Stir in mustard, salt, pepper, and nutmeg if using. Keep warm. Build the Sandwiches:Lightly butter one side of each bread slice. Place two slices butter-side down in a skillet over medium heat. Add 2–3 slices of ham and a generous handful of cheese to each. Top with remaining bread slices, butter-side up. Toast:Grill sandwiches until golden and crisp on both sides and cheese is melted — about 2–3 minutes per side. Top with Sauce and Cheese:Spoon béchamel over each sandwich and sprinkle with extra cheese. Fry the Eggs:In a skillet, fry eggs sunny-side up until whites are set but yolks remain soft. Assemble:Place an egg on top of each sandwich. Serve immediately with a sprinkle of pepper or fresh herbs.
Tips & Substitutions:
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Use Comté or Emmental instead of Gruyère for a different flavor twist.
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Don’t skip the mustard; it adds complexity to the sauce.