When setting aside asparagus pieces before puréeing, choose only the tips, which are usually the most tender part of the stalk and the most attractive.
Hold the asparagus stalk in both hands and press with your thumbs toward the thicker end. Snap off the woody base and discard.
Cook asparagus pieces in boiling water until crisp-tender. Blend all but 1 cup with the reserved cooking water in a blender or food processor to a chunky purée.