Remove the tough green parts from the leek and wash thoroughly to remove dirt. Cut the leek in half lengthwise and then into rough pieces. Sweat on medium in a pot with olive oil, about 10 minutes.
Wash potatoes and cut in small pieces or thin rounds. Add to the leeks, cook on medium until potatoes start to brown. Cover with 400ml vegetable stock, bring to a simmer and cook on low until potatoes are done, about 10-15 minutes depending on the size. Check for seasoning. Add more hot stock if you like the soup thinner. If you like, add finely chopped parsley when the soup is done.
Make the pesto: add all the ingredients to a blender. Blend on high until smooth and set aside.
Ladle soup into bowls, swirl in a spoonful of the pesto, top with pine nuts and enjoy hot.