In a large pot or Dutch oven, melt the butter or heat the olive oil over medium heat.
Add the sliced leeks and cubed potatoes, sprinkle generously with salt and pepper, and cook, stirring occasionally, for about 3-5 minutes until the leeks have softened slightly.
Pour in the warm vegetable stock, bring to a gentle boil, then reduce the heat to a simmer.
Cover and cook for about 20 minutes, or until the potatoes are very tender and easily pierced with a fork.
For a rustic, chunky soup, it's ready to serve now. For a smooth and creamy texture, carefully use an immersion blender to purée the soup directly in the pot until it reaches your desired consistency. You can also transfer it in batches to a regular blender.
Taste and adjust the seasoning with more salt and pepper if needed. Ladle into bowls, garnish with fresh chives, and serve immediately.
