Clone human beings without brains
If a society was medically advanced enough, they could clone human beings without functioning brains. Basically grow them like vegetables in a garden. When the bodies or organs mature, harvest them and those organs would save hundreds or thousands of lives.
Or instead of organ transplants, do a whole body transplant. Just clone a healthy human specimen with a "blank" brain and transfer the neurons of someone's brain into that blank slate, and you basically have immortality.
Of course, there are many other issues that arise with this approach... medical, ethical, religious, financial... just to name a few.