Heat 1 tablespoon sesame oil in a large pot over medium-high heat. Add the carrots, celery and ginger; stir in the scallions, reserving 2 tablespoons scallion greens. Season the vegetables with a pinch each of salt and pepper and cook, stirring occasionally, until the ginger softens, about 2 minutes.
Season the chicken with salt and pepper and add to the pot along with the chicken broth and star anise. Cover and bring to a simmer over high heat, about 5 minutes. Reduce the heat to maintain a simmer, cover and cook until the chicken is just cooked through, 6 to 8 minutes. Remove the chicken with tongs and transfer to a cutting board; let cool slightly, then shred.
Meanwhile, increase the heat under the pot to high. Stir in the spaghetti and 3 cups water; cover and cook, stirring occasionally, until al dente, about 8 minutes. Stir in the wontons and simmer until tender, 1 to 2 minutes. Stir in the spinach and shredded chicken and heat through; season with salt and pepper.
Divide the soup among bowls, discarding the star anise pods. Top with the reserved scallion greens and drizzle with the remaining 1 tablespoon sesame oil.
Don't skip the star anise! It adds great flavor -- plus plenty of antioxidants.