In a blender, combine cucumber, milk, 3/4 cup of yogurt, and lime juice (reserve 1 teaspoon of lime juice) and blend until smooth, about 1 minute.
Add half of each of the following ingredients: avocado and onion, basil, and salt and pepper to taste and blend.
Divide soup evenly between 6 bowls.
In a medium bowl, (do not use a metal bowl) add remaining avocado, onion, basil, and additional salt and pepper to taste, plus 1 teaspoon of lime juice.
Garnish each bowl of soup with avocado mixture and dollop of remaining yogurt and chopped tomatoes.
Serve.