1. Gather all ingredients.
2. In a large saucepan or small Dutch oven, bring the broth, onion, carrots, celery, garlic, and salt to a boil over high heat.
3. Reduce heat to medium-low, simmer until vegetables are tender, about 10 minutes.
4. Using a slotted spoon, transfer vegetables and 1 cup of broth to a blender.
5. Secure lid on blender, remove center piece to allow steam to escape, cover opening with a clean towel, blend until smooth about 30 seconds.
6. Stir pureed mixture back into the soup.
7. Add thyme sprigs, bay leaves, and Parmigiano-Reggiano rind.
8. Bring soup to a boil over medium-high heat.
9. Add pastina pasta, reduce heat to medium-low, simmer stirring occasionally until pasta is tender, 18 to 20 minutes.
10. Add shredded chicken during last 3 minutes of cooking.
11. Remove soup from heat, discard thyme sprigs, bay leaves, and cheese rind.
12. Serve immediately, garnished with fresh parsley and grated Parmigiano-Reggiano cheese.