Melt the butter in a stockpot over medium-low heat. Add the leeks, and cook, covered, until tender, about 15 minutes.
Add the potatoes, stock, and 2 cups water. Bring to a boil; simmer until the potatoes are tender, about 20 minutes. Cool completely; stir in 2 cups of the watercress.
Working in batches, puree the soup in a blender until smooth. Transfer the pureed soup to a large bowl. Season with salt and white pepper. Stir in the half-and-half; chill at least 1 hour. Add the buttermilk just before serving. Adjust seasoning; if necessary, thin the soup with a bit more chicken stock or water to achieve the desired consistency. Garnish with the remaining cup watercress leaves.
