Reserve the garnish: Set out a large bowl and a small bowl. Finely chop ½ cup of the watermelon and place it in the small bowl for garnish. Add half of the remaining watermelon to the large bowl and half to the blender.
Prep the cucumber: Cut off about a quarter of the cucumber, finely chop it, and add it to the small bowl with the watermelon. Cut the rest into 1-inch chunks, adding half to the blender and half to the large bowl.
Prep the tomatoes: Core the tomatoes and cut into 1-inch chunks. Add half to the blender and half to the large bowl.
Prep the bell pepper: Cut the bell pepper into 1-inch chunks. Add half to the blender and half to the large bowl.
Blend the base: To the blender, add the onion, garlic, basil or mint, olive oil, vinegar, salt, and pepper. Starting on low speed and increasing to high, blend until smooth, about 2 minutes.
Add texture: Add the watermelon, cucumber, tomatoes, and bell pepper from the large bowl. Pulse a few times to break them into small pieces, leaving some texture.
Chill the soup: Cover the soup and refrigerate for at least 2 hours, until very cold, to give the flavors time to develop. Add a small pinch of salt to the reserved garnishes and refrigerate alongside the soup.
Garnish and serve: Taste the chilled soup and adjust with more salt or vinegar as needed, since watermelon varies in sweetness. Ladle into bowls and top with the reserved watermelon and cucumber, feta (if using), and a few small basil or mint leaves. Store leftovers in an airtight container in the refrigerator for up to 4 days.
Notes
The flavor of gazpacho changes as it sits. Taste right before serving and adjust the salt and vinegar so it tastes bright, not flat.
This recipe is written for Morton kosher salt. If you use Diamond Crystal, which is less salty by volume, start with a bit more and add to taste.
This soup is even better made in advance. Make it up to 1 day ahead and keep it covered in the refrigerator; the flavors settle as it sits.
For a fuller-bodied, more traditional texture, blend a 1½-ounce piece of day-old baguette into the base in Step 5.
TO STORE: Refrigerate in an airtight container for up to 4 days.
TO MAKE AHEAD: This is one of the rare recipes that improves with time. Prepare it up to 1 day in advance and stir well before serving.