Heat the oil in a very large, deep, lidded saucepan. Add the onion and carrot, and fry over a medium heat for 3–4 minutes, until softened but not coloured.
Drain and rinse the soaked split peas and tip into the saucepan with the onion and carrot. Add the gammon and bay leaves, and cover with cold water. Bring to the boil and skim off any foam that rises to the surface. Turn down the heat, cover with a lid and cook over a low heat for 1 1⁄2 hours, until the peas are tender and the gammon is cooked.
Drain the peas and gammon, reserving the cooking liquid and discarding the bay leaves. Set the gammon aside. Pour the peas into a food processor, add a few spoonfuls of the reserved cooking liquid and blend until smooth and not too thick. Tip into a clean pan and season to taste with salt and freshly ground black pepper. Add the vinegar and butter, beating well until incorporated. Set aside and keep warm.
Preheat the oven to 200°C, fan 180°C, gas 6. Remove the rind from the gammon, leaving a nice layer of fat on the top. Using a sharp knife, score diamond shapes into the fat. Mix together the mustard, sugar and 1 tablespoon hot water to make a thick paste, then pour over the scored gammon. Place on a baking tray and cook in the oven for 20 minutes, until golden brown on top.
Thickly slice the gammon and serve with generous spoonfuls of pease pudding. Great also served with hot parsley sauce.