Combine cucumber and buttermilk in blender. Chop 1/4 of avocado; set aside for salad. Cut remaining avocado into chunks. Add avocado to blender; then add 2 tablespoons red onion and 1 tablespoon basil. Blend until very smooth. Season with salt and pepper. Cover; refrigerate until chilled, about 1 hour.
Mix reserved avocado, remaining 2 tablespoons onion and 1 tablespoon basil, tomato and lime juice in small bowl. (Can be prepared 1 day ahead. Cover soup and tomato salad separately and refrigerate.) Ladle cucumber soup into 4 bowls. Dollop each with 1 tablespoon yogurt; top with tomato salad and serve.
