Season chicken with salt and pepper. In a large pot, heat olive oil over medium heat. Add chicken and cook until browned on all sides. Remove chicken and set aside.
In the same pot, add chopped carrots, celery, and onion. Cook until vegetables are softened, about 5 minutes. Add minced garlic and cook until fragrant – do not burn!
Add a small amount of chicken broth to deglaze the pot, scraping up any browned bits. Add the tomato paste and cook until the color deepens a little.
Return the chicken to the pot. Add the remaining chicken broth, salt, pepper, and dried herbs. Bring to a boil, then reduce heat and simmer for about 30 minutes or until the chicken is cooked.
Take the chicken out of the pot and shred it. Set aside.
Add pasta and cook until al dente, about 8 to 10 minutes (a minute or two less than the directions on the box calls for). Add the fresh spinach a few minutes before the pasta is done, and return the shredded chicken to the pot.
Adjust seasoning with salt and pepper. Serve hot, garnished with plenty of grated Parmesan cheese and garnished with parsley.