Wash out the bung thoroughly and leave it to soak overnight in a bowl of cold water. In the morning turn it inside out and set aside.
Trim the fat from the lungs, liver and heart, wash them under a running tap and transfer to a saucepan of cold water. Bring to the boil and leave to simmer, skimming occasionally, for 1 hour. Scoop out the offal with a slotted spoon, then strain the remaining stock into a jug and set aside.
Once the meat has cooled, chop the liver in half, discard one piece and add the other, along with the lung and heart, to a food processor. Blend for a few seconds until you have a light and even mince.
Transfer the minced pluck to a mixing bowl and tip in the onion, oatmeal and salt. Add a good few grinds of pepper and a splash of the stock to loosen the mixture.
Lay out the bung on the worktop and slice it in two about 40cm from the closed end. You should now have an open-ended tube and a bag of sorts. Setting aside the tube, spoon your mixture into the bag making sure you don't overfill (the stomach will plump up whilst cooking so your raw haggis shouldn't be bulging).
Squeeze any air out of the filled bag (this is important to stop your haggis exploding!), gather up the neck and tightly tie it off with moistened string.
Fill a large saucepan with cold water, bring it to the boil and drop in your haggis. Leave to boil for 3 hours. Serve with mashed swede (neeps) seasoned with plenty of black pepper and boiled and crushed potatoes (tatties).