Heat the olive oil in a large Dutch oven or heavy-based pot over medium heat. Once the oil is shimmering, add the diced onion and garlic, and cook until the onion has softened, about 5 minutes.
Add the chickpeas, the cubed potato, and the tomato paste, and the rosemary sprig. Stir to coat in flavor, about 1 minute.
Then add vegetable broth or hot water, salt, and pepper (1 tsp of salt is fine if you use water or low-sodium broth. But if you use regular broth, it's important to adjust the seasoning at the end rather than at the beginning; see notes below).
Raise the heat and bring the mixture to a boil, then cover the pot, reduce the heat, and cook for 10 minutes until the potatoes are soft.
Turn the heat off, remove the rosemary, and puree only part of the chickpeas with an immersion blender. Alternatively, smash the chickpeas and potatoes with the back of a spoon or a potato masher. Be careful; the liquid is still hot.
Turn the heat on, bring to a light boil, and add the dried pasta.
Let it simmer and stir often (otherwise the pasta will stick to the bottom of the pot). Cook until the pasta is tender but still firm to the bite. You don't need to cover the pot with a lid, but make sure you keep an eye on it. If the soup seems too thick, add a touch of water and thin to the desired consistency.
When it's ready to serve, taste and adjust the season with salt and pepper according to your taste.
Stir freshly chopped parsley, divide into bowls and serve with a drizzle of olive oil, freshly ground black pepper, and grated parmesan or pecorino cheese if you don't keep it vegan. Enjoy!
