Heat oil in a large Dutch oven over medium-high heat. Add the chunky carrots, celery, onion, and garlic to the pot. Sauté for 5 minutes.
Add broth and salt. Simmer for 30 minutes, until the vegetables are very tender.
Use a slotted spoon to transfer the vegetable chunks to a bowl. Set aside to cool for a few minutes (you don’t want to add scorching hot veggies to your plastic blender).
When the veggies have cooled a bit, blend them with 1-2 cups of broth until very smooth. Add the blended mixture back to the pot; stir to incorporate.
Cook pasta according to package directions. Drain and toss with olive oil and a pinch of salt.
Put a scoop of pastina in a shallow bowl. Ladle the steaming-hot broth over the top, then top each bowl with freshly grated Parmesan.
