Prepare the ingredients by washing and peeling the potatoes, then cut them into small cubes. Clean and slice the leeks, ensuring you rinse them to remove any dirt.
In a large pot, melt the butter over medium heat. Add the sliced leeks and a pinch of salt, and sauté for about 5-7 minutes until they become soft and translucent.
Incorporate the cubed potatoes into the pot and stir them with the leeks. Pour in the vegetable broth, increase the heat to bring the mixture to a boil, then reduce to a simmer. Let it cook for 20-25 minutes, or until the potatoes are tender.
Once the potatoes are tender, use an immersion blender to puree the soup until smooth. If using a traditional blender, allow the soup to cool slightly before blending in batches.
Stir in the heavy cream or your chosen alternative and adjust the seasoning with salt and pepper to taste. If desired, reheat gently without boiling.
Ladle the soup into bowls and garnish with chopped chives or fresh herbs for an added touch. Enjoy it warm with crusty bread!