Make the dressing! Add the basil, mint, balsamic, and a pinch of salt to a small food processor or personal cup blender attachment. Pulse a few times until the herbs are finely chopped. Add the olive oil and process until well incorporated. (If you don’t have a small food processor/personal cup blender attachment, no worries! Finely chop the herbs by hand then whisk them with the remaining ingredients for the dressing.) Transfer the vinaigrette to a small bowl and season with freshly ground black pepper to taste. (The dressing should be pesto-like in consistency—thick but still pourable. If it’s too thick, add a splash more olive oil.)
Toast the pistachios. Heat a medium skillet over medium heat. (Don’t put any oil in the skillet!) When the skillet is hot, add the pistachios. Cook for about 3 to 4 minutes, shaking the pan regularly, or until the nuts are lightly brown and fragrant. Transfer to a bowl and cool to room temperature.
In a large bowl, gently combine the watermelon, tomatoes, and feta.
Drizzle 1/3 of the dressing on the base of a platter or shallow serving bowl/dish. Transfer the watermelon, tomatoes, and feta to the serving vessel. Drizzle with the remaining dressing and sprinkle with pistachios.