Position the animal such that it is laying on its side, with the legs sticking out. Start an incision at the base of the skull or base of the tail. Run that incision up or down the back, under the skin, to the opposite end (head to tail, or tail to head). Make an other incision down the back of the rear legs. You now have access to remove the back strap and hind quarter. At the other end, follow the incision around to the base of the head, then down the brisket to the front of the foreleg. You now have access to remove the neck meat and the foreleg. Now that you have removed both legs on one side and the back strap/neck meat on that side, find the rear-most rib on that side. Make an incision at the top of that rib and follow the backbone to the base of the tail. From here you have access to the tenderloin. Because of the position of the animal on it's side, the guts will settled a bit out of the way of your knife blade. You can foster this settling by tipping the carcass up into a bit of a more erect (not laying on its side so much) posture. Once all of the meat you want is removed from this side, flip the animal over onto its other side and repeat the procedure. If you want a shoulder mount, don't make any incisions beyond about the midpoint of the back. With this method you neither have to gut the animal nor hang the animal for quartering.
Quarters can then be left to hang in game bags.
Quote...
"Is it common practice to butcher an Elk on site? What do you do, quarter it and hump it out?"
For some yes and others no. For me, it was a 10 mile hike (w/o horses) from the kill to the truck. So, for me, yes to both questions.
Exposing more surface area of the meat to cool temperatures will increase the rate of cooling. Quartering the animal immediately will decrease the cooling time as you have increased the exposed surface area by removing the skin (insulation) and exposing the surfaces that are bone and muscle junctions while the animal is alive. It will also increase the rate of bleed-out for obvious reasons (more vessels cut).