Combine the leeks, potatoes, milk, and 2 cups water in a 4-quart pot.
Bring to a simmer over medium-high heat. Add 1-1/2 teaspoons salt, reduce the heat to medium low, and simmer until a potato slice falls apart when you poke it with a fork, about 20 minutes. Remove from the heat, stir in the cream, and let cool briefly.
Purée the soup with an immersion blender.
Strain the puréed soup. Let cool to room temperature, stirring occasionally to prevent a skin, and then refrigerate until thoroughly chilled.
Before serving, thin the soup with water or milk if necessary—it should be the consistency of heavy cream. Serve cold, and garnish with chives.