In a small saucepan heat the butter until frothy. Stir in the flour and cook slowly, stirring constantly, for a minute. Add the milk and broth, whisking vigorously and bring the liquid to a boil. Reduce the heat and simmer the sauce for 3 to 5 minutes. Stir in the cheese, remove the sauce from the heat and whisk it just until the cheese has melted. Season to taste with salt and pepper. Preheat the oven to 425 degrees.
Arrange the asparagus in the bottom of a baking dish. Top with sliced eggs and spoon the sauce over the top. Top with bread crumbs mixed with cheese and drizzle melted butter over the top. Bake for 20 minutes or until the cheese sauce is bubbly.