Heat 2 Tbsp oil in a large heavy pot over medium-high heat. Add chorizo or linguica and cook until brown, frequently stirring for about 8 minutes. Transfer sausage to a paper towel lined plate to drain. Keep oil from the sausage in the pot.
Add onions, garlic, chili powder, salt, and pepper to pot and cook until onion is transparent, for about 5 minutes.
Add all of the potatoes and cook until they show signs of softening, stirring frequently, about 12 minutes.
Add the broth and bring to a boil, making sure to scrape up any brown bits from the bottom of the pot. Reduce heat to medium-low, cover, and let simmer until the potatoes are soft, stirring occasionally, about 20 minutes. Using a potato masher, mash some of the potatoes in the pot.
Add sausage to soup in pot. Stir in spinach and simmer just until wilted, about 5 minutes. Stir in remaining 1 Tbsp oil and season with more salt and pepper if needed.