Cut the asparagus into small pieces, including the stems. Spears can be reserved for another dish. In a sauce pan, add 4 cups of water and bring to a boil. Add asparagus stems in for 10 minutes, or until very tender. Drain the water and let the asparagus cool.
In a food processor or blender, add 1/4 cup water plus all of the ingredients except for the oil. Use all of the basil including the stems. Stream in the oil slowly to emulsify. Process until it is smooth and has a pesto consistency.