She sure sounds like a fancy lady. Her sandwich is rich and tasty. Check out the ingredients. I’ve also made something like this without the meat by cutting a hole in a grilled cheese sandwich and then frying an egg inside it while I browned the bread. Keep and brown the leftover “bread circle” if you go that route. It makes an excellent item to mop up the leftover egg yolk.
Back to Madame…you really can’t go wrong with eggs and cheese.
FOR THE SAUCE:
5 Tablespoons Unsalted Butter (divided Use)
3 Tablespoons All-purpose Flour
2 cups Whole Milk
¼ teaspoons Salt
⅛ teaspoons Black Pepper
⅛ teaspoons Freshly Grated Nutmeg
1-⅓ cup Coarsely Grated Gruyere Cheese (divided Use)
8 whole Firm Slices Of White Bread
4 teaspoons Dijon Mustard
½ pounds Thinly Sliced Ham (preferably Black Forest)
4 whole Large Eggs
Melt 3 tablespoons butter in a 1- to 1 1/2-quart heavy saucepan over moderately low heat, then whisk in flour and cook roux, whisking, 3 minutes. Whisk in milk and bring to a boil, whisking constantly. Reduce heat and simmer, whisking occasionally, 5 minutes. Whisk in salt, pepper, nutmeg, and 1/3 cup cheese until cheese is melted. Remove from heat and cover surface directly with a sheet of wax paper.
Spread 1 1/2 tablespoons sauce evenly over each of 4 slices of bread, then sprinkle evenly with remaining cheese (1/4 cup per slice). Spread mustard evenly on remaining 4 bread slices and top with ham, dividing it evenly, then invert onto the cheese-topped bread to form sandwiches.
Lightly oil a 15- by 10-inch shallow baking pan.
Melt 1 tablespoon butter in a 12-inch nonstick skillet over moderately low heat, then cook the sandwiches, turning over once, until golden, 3 to 4 minutes total. Remove from heat and transfer sandwiches to baking pan, then wipe out skillet with paper towels.
Top each sandwich with 1/3 cup sauce, spreading evenly. Broil sandwiches 4 to 5 inches from heat until sauce is bubbling and golden in spots, 2 to 3 minutes. Turn off broiler and transfer pan to lower third of oven to keep sandwiches warm.
Heat remaining tablespoon butter in nonstick skillet over moderate heat until foam subsides, then crack eggs into skillet and season with salt and pepper. Fry eggs, covered, until whites are just set and yolks are still runny, about 3 minutes. Top each sandwich with a fried egg and serve immediately.