In a blender or food processor, combine all ingredients except except a few pieces of strawberry, cucumber, and basil for garnish. Blend until very smooth. If the mixture seems too thick, you can add a tablespoon or two of cold water. Taste and adjust seasoning—it should be a balance of sweet, tart, and savory.
Pour the gazpacho into a container (I use a quart Mason jar), cover, and chill in the refrigerator for at least 1 hour (or up to overnight). Chilling not only makes it refreshing but also allows the flavors to meld and deepen.
Give the gazpacho a good stir. Pour into chilled serving bowls or small glasses. Drizzle each with a little extra olive oil. Garnish with a few pieces of diced strawberry, cucumber, and a basil leaf. Serve cold.