Put leeks in a medium bowl and add cold water. Swirl, then let stand for 5 to 10 minutes. Lift leeks from the water without disturbing the sand/sediment collected in the bowl; allow to drain completely.
Melt the butter in the Cuisinart French Dutch Oven set over medium heat. Add the drained leeks and thyme. Let cook until softened, about 5 to 8 minutes. Add the potatoes, stock, water and 1 teaspoon of the salt and pepper. Cover and bring to a boil over medium-high heat. Once mixture comes to a boil, reduce heat to medium low. Simmer, uncovered, until potatoes are soft, about 12 to 18 minutes.
Using a Cuisinart immersion blender, this soup can be pureed directly in the pan. Be sure the metal blade is completely submerged, and using an up-and-down motion, blend until ingredients are puréed and completely smooth. If an immersion blender is not available, blend in batches in a blender or food processor.
Stir in the salt, pepper and cream or half & half. Taste and adjust seasoning as desired.
Serve immediately or place in the refrigerator until cold for a chilled soup. Garnish with chopped chives.