Toss tomatoes with cucumber, red and green pepper, onion, and garlic in a large bowl. Working in batches, purée the mixture in a blender or food processor. Season to taste with salt, pepper, and sherry vinegar. The amounts needed will vary, depending on your tomatoes and your taste.
Refrigerate overnight. The next day, remove the gazpacho from the refrigerator 45-60 minutes before serving, so it will be cool, but not cold (to avoid the olive oil congealing). Strain the soup, and stir in 1/2 cup extra virgin olive oil for a luxurious texture. The soup may require additional seasoning when chilled, so taste again, and add salt, pepper or vinegar as needed, prior to serving.
To make the basil oil: Purée basil with 1/2 cup extra virgin olive oil and a pinch of salt in the blender.
Serve the soup in chilled bowls drizzled with basil oil.