Heat the butter in a Dutch oven until it melts and the foam subsides. Stir in the leeks. Lower the heat, cover, and simmer until the leeks and onion are translucent.
Add the potatoes, the chicken stock (for Vichyssoise) or water (for French leek-and-potato soup), and the salt and white pepper to taste. Bring to a boil, lower the heat, and simmer, uncovered, for a half hour, or until the potatoes are soft. Mash them with a fork. At this point, you have a traditional French leek-and-potato soup. Serve it hot.
For Vichyssoise, let the soup cool, process in batches, and then stir in the heavy cream. Scatter the chives on top of the soup and serve.