Bring a large pot of salted water to boiling. Add asparagus, cook 5 mins, or until bright green. Drain and place in a bowl of ice water to cool. Drain and set aside.
In a greased 3 quart baking dish, spread half of the bread cubes. top with cheese, onion, chives and half the ham and asparagus. Top with remaining bread.
In a bowl whisk together 4 of the eggs and the milk. Evenly pour over layers in dish. Press down bread pieces into the egg/milk mixture with the back of a spoon. Top with remaining ham and asparagus. Cover, refrigerate until ready to bake.
Bake uncovered in a 325 degree oven 30 minutes. With the back of a wooden spoon, press 6 indentations in top of the strata. Pour a whole egg into each indentations. Back 20-25 minutes more or until an instant read thermometer inserted into center of strata reads 170 degrees and the eggs are set. Let stand for 15 minutes.
Cut into squares to serve. If desired, lightly drizzle with olive oil and sprinkle with salt and cracked black pepper.