Place the tomatoes, balsamic vinegar, avocado oil, and 1 teaspoon of salt into a blender. Puree on high until smooth, then strain through a mesh strainer, and refrigerate until cold, at least 2 hours.
Heat the oil in a skillet over high heat until it begins to smoke. Place the scallops into the hot oil. Cook until the scallops are golden brown on each side, and the center has turned opaque, 1 to 2 minutes per side.
To serve, pour the chilled soup into shallow bowls. Scatter the diced avocado into the soup, and place 4 scallops in each bowl. Sprinkle the soup with ground pepper and torn basil leaves to serve.