Blend diced watermelon, tomatoes, cucumber, garlic, red pepper, green pepper, vinegar, olive oil, salt, paprika, and black pepper. Puree for 2 minutes on high speed till smooth. Your blender's capacity may need batching.
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Transfer gazpacho to a large pitcher or dish. Cover and chill for at least 1 hour, ideally overnight. Longer rests enhance tastes.
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Divide soup into 4 bowls. Add a few garnishes to the soup. Pour extra-virgin olive oil and crush black pepper over.
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