How to Make It
In a large saucepan, combine the chicken stock with the leek tops. Cover and simmer over low heat until the stock is flavorful, about 10 minutes, then strain, pressing on the leeks to extract as much liquid as possible; discard the solids.
In another large saucepan, heat 1/2 tablespoon of the oil. Add the sliced leeks and potatoes and cook over moderate heat, stirring, until the leeks are softened but not browned, 9 to 10 minutes. Add the stock, cover and simmer until the vegetables are tender, about 10 minutes. Stir in the sorrel and cook until wilted, about 2 minutes.
Puree the soup in a food mill or work it through a fine sieve, discarding the fibers and potato skins. Refrigerate until chilled, about 2 hours.
Meanwhile, in a small skillet, heat the remaining 1/2 tablespoon oil. Add the asparagus and cook over moderately high heat until slightly browned, about 5 minutes. Transfer to a plate to cool.
Stir the half-and-half and asparagus into the soup. Season with salt and pepper. Ladle the soup into bowls and serve.