In a country where 300 million people live on less than a dollar a day, Amit Kumar—nicknamed “Dr. Horror” by the Indian media after his arrest last winter for heading an illicit global kidney-transplant ring—had little trouble finding homegrown organ donors. One favorite hunting ground was a strip of restaurants, shops, and hovels near an Islamic shrine, or dargah, in Mahim, a predominantly Muslim precinct of Mumbai. Devotees of the dargah, which attracts people of all faiths, donate money to restaurants to help feed the beggars who cluster there. Last June, walking past one such restaurant whose kitchen extends to...