How to Make It
In a large saucepan, heat 2 tablespoons of the olive oil. Add the onion and garlic and cook over low heat, stirring occasionally, until softened, about 10 minutes. Add the tomatoes, season with salt and pepper and cook over high heat, stirring, until they release their juices, about 4 minutes. Add the chicken stock and basil and bring to a boil. Reduce the heat to low and simmer for 45 minutes. Discard the basil sprigs.
Meanwhile, in a large skillet, heat the remaining 2 tablespoons of olive oil. Add the yellow squash and zucchini and cook over moderately high heat, stirring, until crisp-tender, about 2 minutes. Stir in the sun-dried tomatoes and season with salt and pepper.
Set a coarse strainer over a large bowl. Working in batches, puree the tomato soup in a blender until smooth, then strain it and season with salt and pepper. If serving the tomato soup cold, let cool to room temperature, then chill; let the squash mixture cool to room temperature, too. If serving the soup hot, return it to the saucepan, simmer until heated through and stir in the butter. Add a splash of sherry vinegar to the hot or cold soup and ladle into bowls. Spoon a little of the zucchini mixture in to the center of each soup and serve.