In the bowl of a food processor, combine all ingredients for the vinaigrette and whip until emulsified, about 1 minute. Add salt and pepper to taste. Pour into an airtight jar and keep for up to a week in the fridge.
In a clean food processor, whip feta with olive oil and lemon zest. Spread onto the bottom of a serving platter.
Toss watermelon and arugula with 1 tbsp basil balsamic vinaigrette. Dollop overtop the whipped feta on the platter. Drizzle more vinaigrette, then sprinkle thick salt to finish. Garnish with fresh mint and basil leaves. Eat!