In a large pot, drizzle the olive oil over medium heat.
Add the Chouriço slices and cook, stirring occasionally, until browned, 3-5 minutes.
Use a slotted spoon to remove the sausage to a plate and set aside, keeping all the drippings in the pot.
Add the diced onions. Cook, stirring occasionally, until softened and translucent.
Stir in garlic and cook until fragrant, about 1 minute.
Add the diced potatoes and cover with the 6 cups (1.4 liters) water. Season with salt and white pepper.
Bring to a boil, then decrease heat to medium low.
Simmer until the potatoes are tender, 15-20 minutes. Remove from heat.
Use an immersion blender (or carefully transfer to a blender) to puree the soup.
Bring the soup back to a boil, stir in the kale (some recipes add the reserved Chouriço at this point), reduce the heat to a simmer and cook until the kale is completely tender and vibrant, about 10 minutes. Adjust seasonings to taste.
Serve immediately with the Chouriço (if traditional- with one slice of Chouriço on top and the remaining chouriço on the side).