Croque Madame

Name: Croque Madame
Ingredients: 4 tablespoons salted butter, softened, divided; 1.5 tablespoons all-purpose flour; 1 cup milk; 2 ounces shredded Gruyère cheese; 0.25 teaspoon salt, or to taste; 0.25 teaspoon ground black pepper, or to taste; 0.125 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