What is the proper way to cook a hamburger on a Pit Barrel Cooker that doesn't result in fast cooking taste? I have a Pit Barrel Cooker. I have made hamburgers on it before but I keep running into several issues.  When it comes to anything else, brisket, ribs, chicken, etc. everything turns out great! But hamburgers, even with the provided online videos, my burgers always come out like I just fried them on the stove.  I try to regulate the heat by closing the circular air vent on the front but then my fire dies down too fast and it doesn't seem the burgers are being cooked properly. I use 80% ground beef and my burgers also always shrink up like crazy.  Does anyone have a proper method of using the Pit Barrel Cooker to get slow cooked, bbq tasting burgers?
There is no method because you cannot do it. Ground meats lose moisture rapidly when cooked as much of the fat is not contained within the meat itself, but mixed in. This mixed in fat melts rapidly and runs out. The fat that is marbled within the meat itself is lost faster as the ground strands have much more surface area. Therefore, if you cook a burger low and slow it will end up dry. 

Slow cooking and long roasting is only good when a piece of meat has enough marbled fat so it will still be juicy when cooked, and a low surface area to slow moisture loss (ie it's fairly big). Burger cooking on the other hand is a race to get a good crust and to the right temperature before losing all moisture. 

As for shrinking up this is normal for burgers and you cannot prevent the process from happening. You can, however, shape them flatter and thinner to compensate. I make a flat, thin-ish burger patty, then I use my palm to make the middle slightly thinner than the outside. The result is a burger that has the right shape after cooking. If you start with a thick burger patty you'll more often end up with a meatball shape at the end.