Heat oil in heavy large saucepan over medium-high heat. Add sausage and saute until brown on all sides, about 7 minutes. Transfer to plate using slotted spoon. Add onion to saucepan. Reduce heat to medium and cook until tender, stirring occasionally, about 8 minutes. Add parsnips, rutabaga and carrot and cook 5 minutes, stirring occasionally. Add broths. Bring to boil. Reduce heat and simmer until vegetables are tender, about 40 minutes.
Puree soup in batches in processor or use an immersion bledner to keep some chunks. Return to saucepan. Mix in half and half and thyme. Season with salt and pepper. Add sausage. (Can be prepared 1 day ahead. Cover and chill.) Heat through. Ladle into bowls and serve.