Adjust oven rack to middle position and preheat oven to 400°F (200°C). Line a rimmed baking sheet with aluminum foil and set aside. Using a large knife, trim off stem end of spaghetti squash. Cut squash in half lengthwise, then scrape out and discard seeds. Arrange, cut side up, on prepared baking sheet. Drizzle lightly with olive oil, and rub all over until evenly coated. Season with salt.
Roast squash until its flesh can be pierced with a fork without much resistance but the squash retains an al dente texture, about 35 minutes. Set aside to cool to room temperature.
Meanwhile, combine 1/2 cup olive oil and hazelnuts in a medium skillet. Set over medium heat and cook, stirring frequently, until nuts are lightly toasted and fragrant, about 4 minutes. Stir in cayenne and remove from heat.
When squash has cooled, use a fork to shred flesh into spaghetti-like strands, transferring them to a serving bowl as you go. Discard skins.
Stir in toasted hazelnuts and the oil along with the apple cider vinegar and pomegranate seeds. Season with salt, then stir in mint. Garnish with additional pomegranate seeds and mint leaves and serve right away.