Trim the top and bottom from the leeks, then slice lengthways to the middle and remove the outer layers. Also wipe off any visible dirt. Slice into rounds. Separate the pieces into a colander or large sieve.
Wash the leeks thoroughly under the cold tap.
Peel the potatoes, and cut into large dice.
Melt the butter in a large saucepan over a medium heat. Add the leeks and potatoes. Stir, then reduce the heat to low, and cover the pan with a lid. Allow to cook for about 5 minutes, or until the vegetables are just softening, but not colouring.
Add the stock, and bring the soup to the boil. Season liberally with black pepper. Turn the heat down, cover the pan, and simmer for about 30 minutes or until the vegetables are tender. Test a potato with a knife - it should go in with no resistance when done.
Place the soup into the bowl of a blender or food processor, and blend well. You will likely need to do this in at least two batches. Alternatively, blend with a hand blender in the pan.
Return the soup to the pan and warm through. Taste, and adjust the seasoning if needed.
Ladle the soup into bowls and decorate with a swirl of cream, if liked.
We normally serve the soup hot with crusty bread. But it can also be served well-chilled.
Enjoy!
