This is where most of the meat is won or lost, and it is the part almost nobody is ever taught. Speed and cleanliness in the first hour matter more than anything you do afterwards.
Immediately. Roll the animal on its back, pinch the skin at the base of the sternum and cut upward toward the pelvis, keeping the blade shallow so you never open the stomach or intestines. Two fingers either side of the blade, cutting from the inside out, is the safest way to do it.
Cut around the anus and free it, reach up into the chest and sever the windpipe and oesophagus as high as you can, then roll the whole lot out in one. Prop the cavity open with a stick to get airflow through it. Wipe blood out with a cloth. Do not wash it out with water, which spreads bacteria and keeps the surface wet.
Heat is the enemy and every minute counts. Get the carcass out of the sun, off the ground, and moving air through the cavity. In a warm paddock a gutted animal left flat on the grass will start to sour within hours.
Hang by the hind legs from a gambrel if you can, it makes everything easier. Cut around each leg above the hock and work the hide down. Fallow skin peels relatively easily with fist and knuckle rather than a blade, which is what you want, because every knife stroke on the hide risks putting hair on the meat. Keep the hide falling away from the carcass, not against it.
This is the step people skip and then wonder why their venison is tough. Hang whole or halved at 2 to 4°C for five to ten days, with seven being the sweet spot for fallow. The outside forms a dark dry crust called a pellicle, which is normal and gets trimmed off.
It needs airflow, so never bag it in plastic while ageing. If you haven't got a cool room, quarter it and age the cuts in a fridge instead. If it's too warm to hang safely, don't, just break it down and chill it immediately.
Follow the seams. Muscle groups separate along natural sheets of silver fascia, and if you work with them instead of sawing across them you'll get clean, whole cuts with almost no waste. Quarter into two hindquarters, two shoulders, both backstraps, neck, ribs and trim.
Get the tenderloins out first. They sit inside the cavity along the spine, they're the best two pieces on the animal, and amateurs leave them behind constantly. Eat them fresh that night.
Silverskin, sinew and any fat all come off. Deer fat is not like beef fat, it doesn't render pleasantly and it carries most of the strong flavour people wrongly call "gamey". Take the time here and the eating quality doubles.
Vacuum-seal what deserves it, bag the rest, and label every single package with the cut and the date because in four months you will not remember. Freeze flat so it stacks and thaws evenly. Meat travelling home on ice should stay chilled, not frozen, and go into the freezer at the other end.