In a large saucepan, combine leeks, broth, potato, 2 cups water, and 1 teaspoon coarse salt. Bring to a boil; reduce heat to medium-low, and simmer until vegetables are very tender, 20 to 25 minutes.
Working in batches, puree soup in blender, transferring it to a clean bowl as you work. (To prevent splattering, fill blender only halfway, and allow heat to escape: Remove cap from hole in lid, and cover lid firmly with a dish towel.) Stir cream into pureed soup, and season with salt.
Cover loosely with plastic wrap, and chill until cold, at least 2 hours and up to 2 days. If necessary, thin with water, and season with salt. Serve soup in chilled glasses, garnished with chives.