For the classic cold version, known as vichyssoise just refrigerate the soup until well chilled, then garnish with snipped chives.

Combine the potatoes, leeks, broth, and thyme in a large saucepan.
Cover and bring to a boil over high heat.
Reduce the heat and simmer until the potatoes are tender, about 20 minutes.
Puree the soup in a food processor or blender, in batches if necessary.
Return the soup to the pan, stir in the half-and-half, salt, and pepper, and cook