Add the onion, carrots, celery, garlic, chicken breasts, broth, oregano, and salt to a 6-quart slow cooker.
Cover and cook on LOW for 3 1/2 hours, until the chicken is tender and shreds easily with a fork.
Transfer the chicken to a plate, shred it with two forks, then return the shredded chicken to the slow cooker.
Stir in the orzo, cover, and cook on HIGH for 25 minutes, until the orzo is tender.
Stir in the spinach, lemon zest, and lemon juice, cover, and let stand 5 minutes until the spinach wilts.
Taste and add more salt if needed, then ladle into bowls and top with Parmesan.