Prepare Tahini Dip: Blend all ingredients on low to medium low in a blender until smooth.
Preheat the oven to 425°F, and spray a large rimmed baking sheet with olive oil cooking spray.
Whisk together the umeboshi paste, toasted sesame oil, shoyu, honey, and garlic powder in a small bowl until well combined.
Toss the sliced sweet potato, green beans, and carrot together in a large bowl. Pour the umeboshi mixture over, and stir to cover all of the vegetables with the liquid.
Spread the vegetables out on the baking pan, making sure they don't get too close together or the edges won't crisp. Bake for 15 minutes, then stir; bake for another 15 minutes, and stir again. Bake for 10 more minutes, then remove the pan from the oven.
Top tahini dip with crushed red pepper if you like, then season veggies with additional salt and freshly ground pepper to taste. Serve the dip alongside the roasted vegetables.