Cut the watermelon into cubes, reserving bite-sized pieces for garnish. Hull and quarter the tomatoes, and add them to a separate bowl along with the cucumber, shallots, and Serrano chile. Pour the vinegar over the vegetables and let them soak for about 10 to 15 minutes, stirring once or twice so everything gets immersed. This helps bring out their flavor and brighten them up a bit so they'll stand out more once they all get blended together.
Add the marinated vegetables and the basil leaves to a food processor and pulse until well blended (the mixture should look like a finely chopped salsa). Add three quarters of the watermelon cubes, plus the olive oil, salt, and pepper, and pulse again until completely puréed. Add the last of the watermelon cubes and pulse until the soup has reached your desired consistency (I like a few lingering, smaller chunks in mine, for texture). Taste for seasoning and add more salt and/or pepper if needed. If the watermelon wasn't chilled before making the soup, put the gazpacho in the refrigerator until chilled, about 30 to 60 minutes.
When ready to serve, place the reserved watermelon cubes into four bowls and ladle the gazpacho over them. Sprinkle the crumbled feta over the soup, and finish with a sprinkle of the chopped mint. Serve immediately and enjoy!