Heat 2 tbsp of the oil in a large saucepan. Add the onions, garlic and chorizo, then cook for 5 mins until soft.
Throw in the potatoes and cook for a few mins more. Pour in the stock, season and bring to the boil. Cook everything for 10 mins until the potatoes are on the brink of collapse.
Use a masher to squash the potatoes into the soup, then bring back to the boil. Add the kale and cook for 5 mins until tender.
Ladle the soup into bowls, then serve drizzled with the remaining olive oil.