Trim the tips of the asparagus. Peel the bottom end of the stalks and chop them roughly. Heat one tablespoon oil in a deep pan, add garlic, onion and potato and sauté.
Add bay leaf and continue to sauté for two minutes. Add salt and stir. Add asparagus and stir. Add one cup of water to the first pan and cook till asparagus becomes soft.
Heat one tablespoon oil in another pan, add the asparagus tips and sauté for two minutes. Drain and set aside for garnish.
Heat the remaining oil in the pan. Roughly chop basil leaves and crush them in a mortar with a pestle. Add the hot oil and mix and set aside to cool.
Remove the bay leaf from the asparagus mixture, transfer the asparagus and potatoes into a blender. Add a little cold water and blend into a puree. Let the cooking liquour remain in the pan.
Strain the basil-oil mixture through a muslin cloth and squeeze to remove all the oil. Pour water into a pan and add a little vinegar and heat. Place the ring mould in the centre.
Break an egg in a bowl and gently slide it into the ring mould and poach. Gently scoop the white over the yolk and cook till set. Cook the remaining eggs similarly.
Pour the puree into the cooking liquour remaining in the deep pan. Add milk and stir. Adjust salt and let the soup heat through. Transfer soup into individual soup bowls.
Gently lift the poached eggs from the water and place one over the soup in each bowl.