In a blender, mix the cucumber and buttermilk. Add 1/4 cup avocado, half the red onion, and half the basil. Blend until very smooth.
Season with salt and pepper, then cover and refrigerate until chilled, about 1 hour.
In a small bowl combine the remaining avocado, onion, basil, lime juice, and tomato. Ladle the cucumber soup into the bowls. Dollop each with 1 tablespoon yogurt, and top with the tomato mixture. Refrigerate until time to serve.