Ingredients

• 3 tbsp. unsalted butter
• 3 tbsp. flour
• 2 cups milk
• 12 oz. Gruyère, grated
• ¹⁄₂ cup finely grated Parmesan cheese
• Kosher salt and freshly ground black pepper, to taste
• Freshly grated nutmeg, to taste
• 12 (3/4"-thick) slices pain de mie or Pullman bread, toasted
• 6 tbsp. Dijon mustard
• 12 slices baked ham, thinly sliced
• 2 tbsp. canola oil
• 6 eggs

Instructions

1. Heat butter in a 2-qt. saucepan over medium-high heat. Add flour and cook, whisking, until smooth, about 1 minute. Whisk in milk, and bring to a boil; reduce heat to medium-low and let simmer until slightly reduced and thickened, 6–8 minutes. Add ½ cup grated Gruyère and the Parmesan, and whisk until smooth. Season with salt, pepper, and nutmeg.
2. Heat broiler to high. Place 6 slices bread on a parchment paper-lined baking sheet, and spread 1 tbsp. mustard over each. Top with 2 slices ham and remaining Gruyère. Broil until cheese begins to melt, 1–2 minutes. Top with remaining bread slices, then pour a generous amount of béchamel on top of each sandwich. Broil until cheese sauce is bubbling and evenly browned, about 3–4 minutes.
3. Meanwhile, heat oil in a 12″ nonstick skillet over medium heat. Add eggs, season with salt and pepper, and cook until whites are cooked but yolks are still runny, about 3 minutes. Place an egg on top of each sandwich, and serve hot.