H1B to green card (GC)

Green card: Amazon

The best way to convert H1 visa to GC is to approach a lawyer who specializes in dealing with such issues. There are plenty of good lawyers in New York, Texas and California who have made this their expertise in dealing with medical professionals and converting their visa status. 

You can file for national interest waiver category by yourself but it can be expensive and time-consuming also very tedious. Green cards are usually filed in 2 categories EB1 and EB2, to maximize the chance of being granted one. 
If you chose to seek help of a lawyer, then you will be faced with an expense anywhere from $8000-$12,000 depending on your case. Good luck!