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.

Puree the soup, preferably using a regular blender and working in batches, filling it only half way each time.

Strain the pureed 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.