Preheat oven to 350˚F. Lightly grease an 8x8-inch casserole, and set aside.
Crack the eggs into a large mixing bowl and whisk. Whisk in milk and Dijon mustard, and sprinkle with salt and pepper.
Arrange the bread in the casserole dish, then layer the ham over the bread. Pour the egg mixture over the bread and ham, then allow to rest for 10 minutes. Line the asparagus spears on top of the strata, then sprinkle with the shredded cheese.
Bake for 55-60 minutes, or until the top is golden brown and the eggs are set.