Can my doctor prescribe me something to help speed up my metabolism?

Probably not. Phentermine and Naltrexone potentially increase metabolism by increasing activity of the cart/pomc neurons but this has never been measured. Otherwise the only way to increase metabolism is to over-medicate with thyroid medication which would cause many more problems than it would solve.
Technically, yes. Yes, but unless you actually have a medical disorder that warrants it, it will be more risky than beneficial in the long run. Medications like amphetamines and other stimulants can affect what people refer to as their metabolism but it is only temporary (like 1 day) and your metabolism will go right back to the way it was once you stop the med. And it won't "kickstart" or "jump start" anything!