Heat 3 tablespoons olive oil in a large saucepan over medium heat. Cook and stir onion and garlic in hot oil until the onion is softened, about 3 minutes. Add potatoes; continue to cook, stirring continually, until the potatoes are hot, about 3 minutes more.
Pour chicken stock and water into the saucepan; add chicken soup base. Bring the soup to a boil, reduce heat to medium, and cook at a gentle boil until the potatoes are mushy, about 20 minutes.
Heat a skillet over medium heat. Cook linguica in hot skillet, turning regularly, until no longer pink in the center, about 10 minutes. An instant-read thermometer inserted into the center should read 160 degrees F (70 degrees C). Drain linguica on a plate lined with paper towels and slice.
Pour soup into a blender no more than half full. Cover and hold lid down; pulse a few times before leaving on to blend. Puree in batches until smooth and return to saucepan.
Stir sliced linguica and black pepper into the soup and return to medium heat. Place a cover on the soup and cook at a simmer for 5 minutes.
Stir kale into the soup; simmer until kale is tender and jade green, about 5 minutes more. Stir remaining 1tablespoon olive oil into the soup just before serving.