Heat oil in heavy large saucepan over medium-high heat. Add onion (leek), celery and garlic to saucepan. Reduce heat to medium and cook until tender, stirring occasionally, about 5 minutes. Add rutabaga, turnips, potato and carrots, tomatoes with their juice and two cans broth. Bring to boil, reduce heat and simmer until vegetables are tender, about 40 minutes.
Puree soup in batches in processor or use an immersion blender to keep some chunks. Return to saucepan. Mix in additional broth as desired, add half and half (if using) and thyme. Season with salt and pepper.
Can be prepared 1 day ahead. Cover and chill. Heat through. Ladle into bowls and serve.