Step 1 In a large saucepan or soup pot, heat the oil over medium heat until hot. Stir in the sausage and onion and cook, stirring occasionally, until the sausage browns and the onion softens, about 10 minutes.
Step 2 Stir in the broth, potatoes, kale and water. Bring to a boil over high heat, then reduce to a simmer and cook, partially covered, until the potatoes are very soft (almost falling apart) and the kale is tender, about 30 minutes.
Step 3Use a large fork or whisk to break up some of the potatoes against the side of the pot to thicken the soup. Adjust the liquid as desired, adding more broth or water. Season to taste with salt and pepper. Garnish each serving with a drizzle of olive oil, if desired.