Heat oil in heavy large saucepan over medium-high heat.
Add sausage and cook until golden brown, stirring occasionally, about 3 minutes.
Using slotted spoon, transfer sausage to bowl.
Add onion to drippings in pan and cook until golden brown, stirring often, about 5 minutes.
Add potatoes, broth, garlic, thyme and rosemary.
Bring to boil. Reduce heat, cover and simmer until potatoes are tender, stirring occasionally, about 20 minutes.
Add kale stems and leaves and simmer until tender, stirring occasionally, about 4 minutes.
Mix in sausage. Season to taste with salt and pepper. (Soup can be made 1 day ahead. Cover and refrigerate. Reheat before continuing).
Ladle soup into bowls. Sprinkle with cheese and serve.