Blanching the basil leaves helps preserve their bright-green color. The soup can be made up to one day ahead and stored in the refrigerator; stir it again before serving.
Prepare an ice bath; set aside. Bring a medium saucepan of water to a boil. Add basil, and blanch 1 minute. Transfer basil to ice bath to cool. Place on paper towels; gently squeeze out water.
Transfer basil to the bowl of a food processor fitted with the metal blade; add cucumbers, yogurt, and stock. Process until mixture is smooth. Pass through a fine sieve into a large bowl, pressing on solids to extract liquid; discard solids.
Cover bowl with plastic wrap, and place in refrigerator until soup is well chilled, at least 2 to 3 hours. To serve, ladle soup into shallow bowls, and top each with a scoop of tomato ice.