Gather all ingredients.

Bring broth, onion, carrots, celery, garlic, and salt to a boil in a large saucepan or small Dutch oven over high; reduce heat to medium-low, and simmer until vegetables are softened, about 10 minutes. Using a slotted spoon, transfer vegetables and 1 cup of broth to a blender.

Secure lid on blender, and remove center piece to allow steam to escape. Place a clean towel over opening. Process until smooth, about 30 seconds. Stir blended carrot mixture back into soup.

Add thyme sprigs, bay leaves, and Parmigiano-Reggiano cheese rind. Bring to a boil over medium-high. Add pasta, reduce heat to medium-low, and simmer, stirring occasionally, until pasta is tender, 18 to 20 minutes, adding shredded chicken in the last 3 minutes of cooking.

Remove from heat. Remove and discard thyme sprigs, bay leaves, and Parmigiano-Reggiano cheese rind.

Serve immediately, and garnish with parsley and grated Parmigiano-Reggiano.