In a high-speed blender or food processor, combine the chopped cucumbers, green bell pepper, red onion, and garlic.
Add the fresh dill, mint leaves, lemon juice, Greek yogurt, salt, and pepper to the blender.
Blend on high speed until the mixture is completely smooth and creamy. Scrape down the sides of the blender as needed.
While the blender is running on a low speed, slowly stream in the extra virgin olive oil. This will emulsify the soup, making it even more silky and rich.
Check the consistency. If you prefer a thinner soup, add cold water a tablespoon at a time until you reach your desired texture. Taste and adjust the salt and pepper as needed.
Transfer the gazpacho to an airtight container and refrigerate for at least 2 hours, or until thoroughly chilled. This allows the flavors to meld beautifully.
Pour the chilled gazpacho into bowls. Garnish with a sprinkle of finely diced cucumber, a fresh dill sprig, and a final drizzle of high-quality extra virgin olive oil.