Blanch watercress in pot of boiling salted water 30 seconds. Drain; set aside.
Melt butter in heavy large pot over medium heat. Add potatoes and leeks; sauté 4 minutes. Add 6 cups broth. Bring to boil. Reduce heat; cover partially and simmer until vegetables are tender, about 20 minutes. Add watercress; simmer uncovered 5 minutes. Cool 10 minutes.
Working in batches, puree soup in blender until smooth. Return soup to pot. Mix in cream. Season with salt and pepper. Chill at least 4 hours and up to 1 day. Thin with more broth, if desired. Ladle soup into bowls. Sprinkle with chives.