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If using dried chickpeas, soak them in cold water overnight (about 12 hours); the next day, drain and rinse. If using canned chickpeas, simply drain and rinse.
Heat the extra-virgin olive oil in a large pot over medium heat.
Chop the onion and garlic; slice the carrot into rounds or small cubes.
Add the onion and garlic to the pot and cook, stirring occasionally, until soft and translucent, about 5–7 minutes.
Add the carrot and cook for another 5 minutes, until it just starts to soften.
If using chorizo, add the slices and cook until lightly browned, 2–3 minutes.
Add the chickpeas and diced potato to the pot.
Sprinkle in the sweet paprika and cumin (if using), add the bay leaf, and stir well to coat everything in spices.
Pour in the vegetable broth or water until the ingredients are just covered, then bring to a boil.
Reduce the heat to medium-low, cover, and simmer: 1½–2 hours for soaked chickpeas or 20–30 minutes for canned, until the chickpeas and potatoes are tender.
Season with salt and freshly ground black pepper to taste. For a thicker texture, mash a few chickpeas and potato pieces against the side of the pot.
Let the stew rest for 5–10 minutes before serving. Serve with crusty bread, if you like.