In a large soup pot, mix together the chicken, onion, celery, carrots, potatoes, turnips, chicken broth, water, and thyme and bring to a simmer over medium-low heat. Reduce heat to low and simmer – the soup will barely be bubbling – for 1 to 1-1/2 hours.
Before serving, add the cornstarch mixture, while stirring the soup, until the soup thickens. Season with brown sugar, balsamic vinegar, salt and pepper and pour over large toasted croutons. Serves 6 to 8.