In a pot over medium heat, bring the broth, carrots, celery, and onion to a simmer. Add in the chicken and cover with a lid. Simmer until the chicken reaches an internal temperature of 160°F using a thermometer.
Remove the chicken to a cutting board and shred apart using 2 forks pulling in opposite directions. Then return the chicken to the pot and return to a simmer. Season with salt to taste.
Add pastina and cover with a lid. Stir occasionally (especially at beginning as pastina may stick to the bottom) and cook until pastina is tender, about 8-10 minutes. Once the pastina is cooked, remove from the heat (stir in more broth if needed).
Stir in the spinach until wilted, then stir in the parmesan cheese. Taste and adjust the seasoning with salt or more parmesan if needed.
Ladle into bowls and garnish with Parmesan cheese, if desired. Serve with a roll.