Add the garlic, carrots, celery, onion, salt, and black pepper to a medium soup pot or large saucepan and cover with four cups of low-sodium chicken broth.
Bring to a boil over high heat, then reduce the heat to medium-low, cover and gently simmer for 30 minutes.
Remove from the heat and let the broth and veggies cool for 30 minutes. Use kitchen tongs or a slotted spoon to strain the veggies from the broth, transferring them to the bowl of a large blender along with 2 cups of the broth (reserve the rest of the broth in the pot).
Blend until smooth, then pour it back into the pot and stir (optional: strain the blended veggies and broth through a fine mesh strainer or cheesecloth, if desired). Season with additional salt and black pepper, to taste.
Increase the heat back to high and return the broth to a boil. Once boiling, reduce the heat to medium-low and cover with a lid. Simmer for approximately 30 minutes or until it develops a deep orange color and has reduced a little bit. (If it reduces too much, pour in a cup of broth or water at a time until it’s to your liking.)
Meanwhile, cook the pastina in a separate pot of boiling water until al dente. Drain and rinse well.
Just before serving, stir the half cup of grated parmesan cheese into the broth.
To serve, spoon desired amount of pastina pasta into a bowl, then pour the broth over the top. Serve with additional parmesan cheese if desired.