Using a knife, chop off the woody ends (about a couple of centimeters/1 inch) and the heads (set these aside, reserve for garnish).
With a potato peeler, peel off the outer part of the asparagus then cut the 'body' in small pieces.
Bring a pot of water to a boil and cook the heads for about 3 minutes. Drain (but save the water) and season them with butter/oil and salt (also lemon juice, if you want). Set aside.
To make the soup:
Rinse and peel your two small potatoes. Chop them into small chunks.
In a pan, stir-fry a small shallot with a drizzle of oil, add the asparagus (except for the heads) and the chopped potatoes. Cook for about 5 minutes, keep stirring.
Add 250 gr of the water you used to boil the asparagus, cover and cook until the potatoes and asparagus are tender.
Add salt, pepper, ginger powder and milk and blend using a food processor.
Garnish with sesame seeds and lemon zest (optional).