In a large bowl, combine the melons, strawberries, vinegar, olive oil, chile pepper, if using, salt, and black pepper. Set aside and marinade for 30 minutes, or longer.
For the whipped feta, combine feta, yogurt, and olive oil in a blender or food processor and pulse until smooth and creamy. Set aside.
For the basil vinaigrette, combine basil leaves, shallots, garlic, vinegar, salt, red pepper flakes, and olive oil, in a blender or food processor and blend until smooth. Season to taste, if needed.
To assemble the salad, spread the whipped feta on the bottom of a flat plate. Top with the fruit and spoon as much vinaigrette desired over the fruits and the feta. Garnish with pistachios, basil, and mint. Enjoy!