Heat a glug of olive oil in a large pot. Add sliced chorizo and onion, cooking until onion is soft and chorizo has released some of its flavorful red oil. Add minced garlic and soften.
Stir in chopped potatoes and pour in the stock. Bring the mixture to a boil, then reduce head to low, cover and simmer until potatoes are tender.
Once potatoes are soft, use a potato masher or immersion blender to partially mash the potatoes into the soup.
Add the kale to the pot. Stir and simmer until the kale is wilted and tender.
Season with salt and pepper to taste. To serve, ladle into bowls, drizzle with olive oil and finish with a side of crusty bread.