In a large saucepan, saute the onion and garlic in 3 Tablespoons of oil for about 3 minutes. Add the potatoes and saute, stirring constantly, for 2-3 minutes. Add water, cover, and boil gently over medium heat for 20 minutes, until the potatoes are mushy. Meanwhile, fry the sausage in a skillet over low heat for about 10 minutes, until most of the fat has drained out. Drain well and reserve.
When the potatoes are soft, remove from the heat and either mash them in the pan with a masher or puree them--then add the sausage, salt, and pepper. Return to medium heat, cover, and simmer for 5 more minutes.
When ready to serve, add the shreds of greens and simmer uncovered for about 5 minutes--until they are tender and the color of jade. Mix in the last Tablespoon of oil, taste for seasoning, and ladle into large flat soup plates.