1. In a large soup pot, heat the oil over medium heat. Saute the leeks for 5 to 7 minutes or until softened. Add the squash and saute for another 5 minutes or until lightly browned. Add the garlic and cook for another minute. Season with salt and pepper.
2. Add the stock and cook the soup, covered, for about 15 minutes, or until the squash is tender. Process the soup with a hand blender in the pot or in the blender until pureed.
3. Pour into a bowl. Place in the refrigerator and chill for at least 4 hours.
4. Right before serving, add the lemon juice, basil, chives and buttermilk. Season with salt and pepper. Taste for seasoning. Chilled soups usually need more seasoning, because the chilling process reduces the flavor. You can always add some extra liquid to re-create the proper texture and thickness.
5. To serve, ladle into soup bowls and garnish with shredded basil.