Heat olive oil to a large pot over medium heat. Add the onion, garlic, carrots, celery, Italian seasoning, and bay leaf. Cook, stirring often until vegetables are tender, about 7-8 minutes.
Add 8 cups of chicken stock, Better Than Bouillon, rosemary sprig, salt, and pepper then bring to a boil. Reduce heat and simmer, uncovered, until the veggies are tender (about 10-12 minutes).
Stir in the pasta and cooked chicken. Simmer just until the pasta is al dente, then remove the pot from the heat. The pasta will continue to soften slightly in the hot broth. Adjust seasoning if needed.
To serve, ladle soup into each bowl, then top with a generous pinch of shaved Parm.