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 tsp. 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, preferably using a regular blender and working in batches, filling it only half way each time.
Strain the puréed soup through a fine sieve. Let cool to room temperature, stirring occasionally (stirring prevents a skin from forming), and then refrigerate until thoroughly chilled.
Before serving, thin the soup with water if necessary—it should be the consistency of heavy cream. Season to taste with salt. Serve cold in chilled bowls, garnished with the chives.