1. Roast the kabocha squash. Preheat the oven to 425°F (220°C). Rinse the outside of the squash and cut it in half. Place both halves cut-side up in the oven, and roast for about 35 minutes, until the tip of a knife pierces through the interior easily. Remove the seeds with a spoon and discard; leave the peel on. Slice the squash into 1-inch U-shaped wedges, then cut into 2-inch chunks.
2. Stir-fry the aromatics. In a wok or large saucepan over medium heat, heat the oil until shimmering. Stir-fry the ginger and scallion whites until fragrant and the ginger is starting to color, about 2 minutes. Add the garlic and stir-fry for another 30 seconds. Pour in the water, then add the sugar, soy sauce or tamari, and mirin. Stir until the sugar is dissolved and the liquid is starting to bubble. Add the rice wine vinegar and gochugaru chili powder. Continue to cook for 2 more minutes until the liquid is reduced by a third and thickened.
3. Add the kabocha squash to the sauce. Fold in the roasted kabocha squash pieces gently into the sauce until they are thoroughly coated. Simmer for 1 more minute, then remove from heat.
4. Serve. Transfer to a serving dish along with any remaining glaze. Drizzle with the sesame oil and garnish with the thinly sliced scallion greens and toasted sesame seeds. Serve immediately.

Store the leftovers in a covered container in the refrigerator for 3 to 4 days. This dish can be enjoyed chilled as well.