Heat the oil in a medium nonstick pot over medium heat. Add the onion, celery, carrot and garlic and sauté until soft, 4 to 5 minutes. Add the chicken, broth, bay leaves and 1/8 teaspoon black pepper and bring to a boil. Cover and reduce to medium-low until the chicken shreds easily, about 25 minutes.
Discard the bay leaves, coarsely shred the chicken with two forks and return to the soup, add the pasta and cook according to package directions, adding the baby spinach at the last minute to wilt.