Heat the butter in a saucepan, add the chopped fennel, spring onions and potato, toss in the butter then cover and fry gently for 10 minutes, stirring occasionally until softened but not browned.
Pour in the stock, season and bring to the boil. Cover and simmer for 15 minutes until the vegetables are just tender and still tinged green.
Allow the soup to cool slightly, then purée in batches in a blender or food processor until smooth. Pour the purée through a fine sieve back into the saucepan, then press the coarser pieces of fennel through the sieve using the back of a ladle. Mix in the milk and cream, then taste and adjust the seasoning if needed. Chill well.
Ladle the soup into small bowls or glasses half filled with ice and garnish with the reserved green feathery tops, snipped into small pieces.