Croque Madame Sandwich

Ingredients (13)

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