Discard asparagus ends. In a small saucepan cook the reserved asparagus tips in water, uncovered, over high heat for about 3 minutes or until they are crisp and tender. Transfer asparagus tips with slotted spoon to a colander, reserving cooking liquid, and rinse under cold water to stop cooking. Drain the tips well.
In a 4-quart heavy sauce pan cook onions in O Extra Virgin Olive Oil over moderate heat, stirring until tender. Salt and pepper. Add asparagus stalk pieces, broth, reserved cooking liquid and simmer covered for about 15 minutes. Purée soup in a blender in small batches and place in large glass bowl. Adjust salt and pepper if desired. Stir and chill.
Divide soup among six bowls and add asparagus tips decoratively. Drizzle each bowl with O Fresh Basil Olive Oil and serve. Delicious.