Peel the Jerusalem artichokes, putting the peeled roots into a bowl with the milk and enough water to cover them – this stops them discolouring. Pour into a pan (top up the water, if necessary), add the bay leaves and boil until soft, then drain and rinse.
Put the 25ml of olive oil into a pan on a medium-low heat and gently fry the garlic, shallots and jamón – you don’t want them to colour. Add the chicken stock and cook until it reduces by half, then add the artichokes, season, and continue to cook. When there is hardly any liquid in the pan and almost all the stock has been absorbed by the artichokes, the mixture is ready to blend. Add the additional 40ml of olive oil when blending, to emulsify – this gives the purée a creamier texture. Transfer to a saucepan and keep warm.
Put 1cm of oil into a pan on a medium heat, and crack in the eggs, 2 at a time. Fry until they are crispy at the edges – I like to spoon some of the hot oil over the yolks, so that they cook slightly but are still runny inside. I also lift up the white around the yolks four times so that the oil can better cook it. When the eggs are golden and crispy at the edges, remove them with a slotted spoon to a paper-lined plate. Season with a tablespoon of olive oil, salt and pepper, then cook the next 2 eggs in the same way.
Put the artichoke purée on plates, top with the eggs and shave over the truffle, using a truffle mandoline or the small, lumpy side of a grater. Serve with good bread.