A croque-madame is a variation on the croque-monsieur, a French twist on grilled ham and cheese. Instead of dipping the sandwich in egg before sautéeing in butter (as you would for a croque-monsieur), grill the sandwich first and top with homemade Mornay Sauce, cheese, and sunny-side up egg.
2 tablespoons melted butter
4 slices sourdough or white sandwich bread
4 slices good-quality cooked ham
1 1/2 tablespoons Dijon mustard (optional)
1 cup grated Gruyère or Emmentaler cheese, divided
Preheat broiler. Heat a large ovenproof skillet over medium-high heat. Butter bread on all sides, and top 2 of the slices with ham and, if desired, a smear of Dijon. Top with 1/2 cup cheese, and cover with remaining bread slices. Place sandwiches in skillet, pressing gently with back of spatula. Let cook about 1 to 2 minutes or until bottom is lightly golden.
Top sandwiches in skillet with Mornay Sauce and remaining 1/2 cup cheese. Place skillet in oven, and broil 2 to 3 minutes or until golden and bubbly, taking care not to burn. Top with egg, and serve immediately.