Anthony J Sargeant says that you should not waste the trimmings when you snap off the base of the asparagus.
First trim the scales off the asparagus (they are very fibrous and it is best to remove them from the whole of the stalk up to about an inch or two of the top of the stalk). Do this before snapping off the bottom section of the stalk. But retain this trimming and using a sharp knife chop thin cress sections down towards the base. If you feel any woody resistance discard that very last section (often there is none). Sweat off some onion in butter (but do not brown) then add the thinly sliced asparagus sections and cook until tender. Add some chicken stock cook-off then liquidise, with seasoning added as required, until smooth. The result is a smooth asparagus soup which can be garnished with chives and perhaps a swirl of double cream. It is delicious and sweet.