Start by cleaning the leeks thoroughly. Trim off the dark green parts and slice the white and light green parts into thin rounds.
In a large pot, melt the butter over medium heat. Add the sliced leeks and sauté until they become soft and translucent, stirring occasionally.
Peel the potatoes and cut them into small cubes. Add the potatoes to the pot with the leeks and cook for a few minutes, allowing them to absorb the flavors.
Pour in the chicken or vegetable stock, season with salt and pepper, and bring the mixture to a boil. Reduce the heat to low and simmer until the potatoes are tender and can be easily mashed.
Using an immersion blender or a regular blender, puree the soup until smooth and creamy. If using a regular blender, be sure to let the soup cool slightly before blending, and blend in batches if necessary.
Return the soup to the pot and stir in the heavy cream. Adjust the seasoning if needed.
Transfer the soup to a container and refrigerate for at least a few hours, or until it is well chilled.
When ready to serve, garnish the Vichyssoise with chopped chives and a drizzle of olive oil if desired. Enjoy the soup chilled for the ultimate refreshing experience.