Bring a large pot of salted water to boiling. Add asparagus; cook 5 minutes or until bright green.
In a greased 3-quart baking dish spread half the bread cubes. Top with cheese, onion, chives and half the ham and asparagus. Top with remaining bread.
In a bowl whisk together four 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 with plastic wrap, then chill for two hours.
Bake, uncovered, in a 325 degrees F oven for 30 minutes. With the back of a wooden spoon, press 6 indentations in top of strata. Pour a whole egg into each indentation. Bake 20 to 25 minutes more or until an instant-read thermometer inserted in center of strata registers 170 degrees F and eggs are set.
Let stand 15 minutes.
Cut into squares to serve. If desired, drizzle lightly with olive oil and sprinkle with salt and cracked black pepper.
MAKE-AHEAD TIP:
Prepare as directed (through step 3). Cover; refrigerate up to 12 hours. To serve, preheat oven to 325 degrees F, then bake for 30 minutes. Remove from oven, then press 6 indentations in top of strata. Pour a whole egg into each indentation. Bake 20 to 25 minutes more or until an instant-read thermometer inserted in center of strata registers 170 degrees F and eggs are set. Let stand 15 minutes.