Cut the leek lengthwise and chop into small pieces. Chop the potato into small pieces.
In a large saucepan, heat butter on medium-high heat. Add the potato and leek in the pan and sauté until they are well coated with butter and the leek gets wilted.
Add vegetable stock. Cover with a lid and bring it to a boil over high heat. Keep simmering until the potato is soft.
When a skewer pierces through the potato easily, turn off the heat. Pour the soup into a blender.
Add the milk and mix.
Pour the soup back into the saucepan. Add salt and pepper. Reheat when you serve. Do not boil the soup. Garnish with parsley and serve immediately.
To Store
You can keep the leftovers in an airtight container and store in the refrigerator for up to 2 days.