Combine sugar, vinegar, ginger, salt, and cayenne pepper in a small saucepan. Heat over medium heat, whisking until sugar is dissolved. Transfer to a jar with a tight-fitting lid (like a Mason jar), add vegetable oil, olive oil, sesame seeds, and sesame oil and shake until combined. Chill in the refrigerator until ready to serve.

Arrange watermelon and apricots on a large platter. Using a vegetable peeler, peel the cucumber into long ribbons, stopping when you get to the seeds in the middle. Turn the cucumber and start from the other side, stopping again when you get to the seeds in the middle. Repeat until all your left with is the core of seeds. Take one cucumber ribbon, twist it into a tight spiral and tuck it in among the watermelon. Repeat until all the cucumber ribbons are gone. Sprinkle the shiso leaves over the salad, drizzle about half of the dressing over the salad. Serve with remaining dressing on the side.