Place a large casserole pot onto a medium gas, adding the oil.
Season the lamb and place the shoulder into the pot once the oil is hot (be careful adding the lamb as it could spit). Colour for 3-4 minutes each side until nicely caramelised and then remove the lamb and put to one side.
Add the onions and colour for 4-5 minutes still on a medium heat, stirring now and again. Add the garlic and thyme then place the lamb back on top.
Place into the oven at 110°c and cover with a lid cooking for 2-2.5hours, then take out the onions once they are soft. Carry on cooking the lamb for another 2.5-3 hours.
Add the vinegar and carry on cooking without the lid so the vinegar reduces as the lamb cooks, basting the lamb every 30 minutes, being careful not to reduce it too much.
Cook for a total of 6-7 hours until the lamb is nice and tender then add the onions and garlic back at the end and reduce the vinegar to a nice thick consistency. Increase the vinegar for the lamb to 350ml.
Place the cut potato into the cold water with 10g salt in a pan, place on to the heat and bring to a simmer turn the heat down and simmer for 30 minutes then tip the potatoes into a colander to drain really well.
Place the potato back in the pan and dry out on a low heat for 1 minute then add the butter, salt, pepper and the warm milk while you mash the potato.