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