First make the stock. Put the 4 onion halves into a pan on a medium-low heat with no oil and burn the cut side of the onions until black – this will take about 20 minutes.
Put all the meat into a very large pan, cover with cold water and bring to the boil. Simmer for 10–12 minutes – all the impurities will rise to the surface. Drain, then rinse the meat and the pan. Return the meat to the pan and cover with 5 litres of fresh water. Wrap the chickpeas in muslin, tie together and add to the pan. Add the wine.
Tie the parsley and bay together. Tie the leeks and carrots together separately. Add to the pot with the garlic, cut in half horizontally, and the burnt onions. There should be enough space for everything to float. Cook very gently on a low heat for 3 hours minimum.
Remove the meat from the pot, and the vegetables separately. The garlic, parsley, bay leaf, beef knee joint, ham bones and onions can be discarded. Next, you need to strain the stock through some muslin. The simplest way to do this is to tie the muslin over the top of a clean saucepan – make sure you leave enough slack so that it doesn’t buckle. Gently pour the stock through the muslin and set aside to cool.
Pick the meat off the bones. Chop the vegetables into small cubes.
Put 500ml of the cooled stock into a bowl with the egg whites and whisk together. Pour back into the pan of stock, stir a little, then leave it alone! Put the pan on a very low heat: you need it to be really gentle or the egg whites will start to cook. The egg whites will start to rise and form a crust, bringing all the impurities up with them – when the stock reaches this point, it’s ready. Carefully remove the crust with a ladle.
Gently pour the stock through the muslin again – you should now have a clear stock. For 4 people, you will need around 800ml of stock, but you can freeze any extra for use another time.
Fill a saucepan with water and bring to the boil. Add vinegar and salt. Crack a duck egg into a small ramekin or glass, then gently pour the egg into the boiling water and poach for 1½–2 minutes. Remove to a plate with a slotted spoon and repeat until you have poached all the eggs. Drizzle over a little olive oil and sprinkle with smoked paprika.
Warm up the stock. Divide the vegetables and meat between four bowls, then pour over the stock, add a semi-poached egg to each bowl, and leave for 30–40 seconds to continue cooking before serving.