In a small bowl, combine the vinegars, salt, pepper, and whisk until salt is dissolved. Slowly whisk in the olive oil, a few drops at a time. Add in the chopped basil, taste, and adjust seasonings.
In a large bowl, combine the melon, arugula/spinach, onion, and feta. Pour the dressing over the melon mixture and toss gently until everything is coated and evenly mixed.
To serve, divide salad among individual plates and garnish with slivered almonds and extra basil leaves.