Croque Madame Sandwich

Details

Ingredients

• 4 tablespoons salted butter, softened, divided
• 1 ½ tablespoons all-purpose flour
• 1 cup milk
• 2 ounces grated Gruyere cheese
• ¼ teaspoon salt, or to taste
• ¼ teaspoon ground black pepper, or to taste
• ⅛ teaspoon ground nutmeg
• 4 slices white bread, lightly toasted
• 2 teaspoons Dijon mustard
• 2 thin slices Black Forest ham
• 2 large eggs
• 1 tablespoon water
• 1 teaspoon chopped fresh dill, or to taste