How Many Heart Transplant Surgeons Are There?

You can call a Heart/Lung transplant surgeon a Cardiothoracic surgeon for simplicity's sake.

Quite frankly, there are very few.  Not sure on the numbers, but let me tell you why are there are very few:

1) 4 years of College with top notch grades - Required.
2) 4 years of Medical school with top notch USMLE scores - Required.
3) 5 years of a general surgery residency - Required.
4) ~2 years of Cardiothoracic surgery residency - Required.
5) 1-2 years of Cardiothoracic surgery fellowship (i.e. you can specialize in adult or child cardiothoracic surgery or heart transplants) - Required for heart transplanting.
6) Beginning with step 2, you will suffer grueling grueling GRUELING study nights/late calls/horrible residents that treat you like trash/and then working ~80hrs/week averaging ~$12/hour.
7) Cardiothoracic residencies are VERY COMPETITIVE and only a handful of people make it.  For example, at UCLA, about 2 doctors per year are accepted into the Cardiothoracic residency program.  

Beginning with step 3 above, you'll start off making ~35K and it'll go up as the years go by.