Bring a pot of salted water to a boil. Add the asparagus and cook for 1 to 2 minutes, or until bright green. Immediately immerse in iced water to stop the cooking process. Set aside.

Spray a 3-quart baking dish with cooking spray. Then spread half of the French bread cubes in the bottom of the baking dish. Top with shredded cheese, green onion, and half of the ham and half of the asparagus. Top with remaining bread cubes.

In a bowl, whisk together 4 eggs and the milk. Pour evenly over the strata, pressing the bread cubes down into the liquid with the back of a spoon. Top with remaining ham and asparagus. Cover and refrigerate until ready to bake; 6 hours to 1 day ahead is best.

Baking the Strata: Preheat the oven to 325 degrees F. Uncover the strata and bake for 30 minutes. Remove from the oven and with the back of a spoon, make 6 wells or indentations for the eggs. Carefully pour a whole egg into each of the 6 wells. Return the strata to the oven and bake 20 to 25 minutes more or until a thermometer inserted in the center registers 170 degrees F and the eggs are set.

Let stand 15 minutes before cutting into squares. If desired drizzle with olive oil and sprinkle with fresh chopped parsley, salt, and black pepper.