Chop the carrots, celery and onion into small pieces. Heat the olive oil in a large soup pot or a dutch oven. Add the vegetables and saute until they begin to soften, about 5 minutes.
Add the rinsed rice, poultry seasoning, thyme and bay leaf. Saute for 2 minutes. Add the broth and bring to a boil on high heat. Once boiling, lower heat to medium-high and let cook for approximately 30 minutes.
Stir in the chopped chicken and continue to cook until the rice is fully cooked, about 15 minutes. Add more water if needed.
Add the spinach, parsley and milk and let simmer for 5 more minutes. Remove from heat and season with salt and pepper as needed. Remove bay leaf before serving.