Peel the vegetables; quarter the onion, roughly chop the celery and carrots and slice off the top of the head of garlic (or wash well and leave the peels on)
In a large pot, combine water, stock, vegetables, garlic, bay leaves and Parmesan rind
Season with 1 tsp salt and bring to a boil, then reduce to a simmer and cover with a lid
Allow to simmer for 30-40 minutes until the vegetables are soft and the carrots can be easily pierced with a fork; remove the Parmesan rind
Blend the vegetables using either an immersion blender directly in the pot or by transferring the vegetables and 1-2 cups of liquid to a stand blender, then returning the mixture to the pot
Leave the pureed vegetable broth over low heat so it stays warm
In a medium saucepan, melt 1 tbsp of butter and toast the pastina for 2-3 minutes (see Note 2)
Pour in enough of the warm broth to cover the pastina and the remaining tsp of salt
Bring to a low boil; keep stirring so it doesn't stick to the bottom of the pot
Add a ladle of broth at a time and continue cooking until the desired consistency is reached and the pastina is al dente (about 2-3 cups of broth; see Note 4)
Add the Parmesan cheese and freshly cracked black pepper and stir, then taste and season with additional salt if needed
Stir in the remaining tbsp of butter (see Note 5)
Serve topped with additional Parmesan cheese, freshly cracked black pepper, lemon juice or extra virgin olive oil if desired