Set a large heavy pot or Dutch oven on the stove over medium heat and add the olive oil.

Add the chopped onion and celery to the pot. Cook for 4 to 5 minutes, stirring often, until the vegetables look softened and the onion is no longer raw-looking.

Add the potato pieces and 1 teaspoon of the kosher salt to the pot. Stir well so the potatoes are coated with the oil.

Cook the potatoes for 3 minutes, stirring a few times, so the edges start to lose their raw look.

Add the sliced garlic to the pot. Cook for 30 seconds, stirring, until the garlic smells fragrant.

Add the chopped rosemary and black pepper to the pot. Stir for 30 seconds to warm them in the oil.

Pour the vegetable broth into the pot. Stir and scrape the bottom of the pot with a spoon so nothing is stuck there.

Turn the heat to high and bring the broth to a full boil.

Add the ditalini pasta to the boiling broth and stir right away so it does not clump on the bottom.

Lower the heat to medium-low so the liquid stays at a steady gentle bubble. Cook for 8 minutes, stirring every 1 to 2 minutes to keep the pasta from sticking.

Use the back of your spoon to press a few softened potato pieces against the side of the pot. This helps thicken the broth.

Keep cooking for 4 to 6 minutes more, stirring often, until the pasta is tender and the broth has thickened to a loose, spoonable texture. If the pot looks too thick before the pasta is ready, add a small splash of water.