Put the smothered cabbage and broth in a soup pot and bring to a boil over medium heat.
Add the rice and cook uncovered at a slow, steady boil, stirring from time to time, until the rice is tender but still firm, about 20 minutes. If the soup thickens too much, loosen it with broth (or water if you are not using homemade broth).
When the rice is done, stir in the butter and Parmesan, mixing well. Taste and adjust salt, add a few grindings of pepper, then ladle into plates and let it rest a few minutes before serving.