Stir in the flour and cook gently, stirring, for 1-2 minutes.
Gradually add in the warmed milk, stirring well as you do so. Cook, stirring, until the mixture has thickened into a white sauce.
Add the nutmeg and Cheddar cheese, stirring well. Cook gently until the cheese has melted.
Preheat the broiler.
Broil the ham steaks until cooked on one side. Turn over the steaks.
Top the uncooked side of the ham steaks with the hard-boiled egg quarters. Spoon over the cheese sauce, sprinkle with the cayenne and Parmesan and cook briefly under the broiler until glazed. Serve at once.
Excess sauce can be stored away in a air-tight container for up to two days.