Peel and trim the asparagus, reserving the trimmings. In a large saucepan bring stock or broth to a simmer, add trimmings and let stand 15 minutes.
Rinse spears, pat dry and cut into 1 1/2 pieces, reserving tips. Strain stock or broth into a large saucepan and add the asparagus stalks, shallot, potato and salt and pepper to taste. Bring the liquid to a boil over high heat and simmer the mixture, stirring occasionally, for 25 minutes, or until asparagus is tender.
Meanwhile, in a saucepan of boiling salted water blanch the asparagus tips until just tender, drain and refresh.
In a food processor or blender puree the soup in batches until smooth, correct seasoning and return to saucepan. Heat until hot.
To serve: ladle soup into bowls and garnish with reserved asparagus tips, yogurt, lemon peel and dill.
