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.
The recipe says to see the note below regarding cleaning the leeks but there is no note below