Cut the spaghetti squash in half lengthwise (see notes). Scoop out the seeds and pulp. Place squash, cut sides down, into a microwavable dish. Add ½ cup of water to the dish and microwave for 10 minutes on full power, or until soft.
Remove from microwave and let cool for 5 minutes. Once cool enough to handle, use a fork to scrape out the flesh while maintaining the stringiness of the squash. Set aside until ready to use.
Heat olive oil over medium heat in a large soup pot.
Add onion, carrots, and celery and sauté for 5 minutes or until starting to soften.
Add garlic, thyme, salt, and pepper. Continue cooking over medium heat, stirring occasionally for one minute.
Add chicken broth and chicken. Bring to a gentle simmer. Cover and cook for 15 minutes, or until chicken is cooked through.
Remove chicken and shred or chop into bite-sized pieces.
Add chicken and arugula or spinach (if your squash is done you can add it here) and simmer gently for 10 minutes.
Add chicken back to the soup pot along with the squash and spinach or arugula. Simmer for 5 more minutes.
Taste and adjust seasoning as needed.