Place ham hock into large saucepan of cold water with the carrot, celery, sliced onion and bay leaves Simmer for 4-5 hours, topping up the water as needed, until the meat pulls easily away from the bone. Lift out the hock and when it is cool enough to handle, pull it apart and roughly shred the meat, discarding the fat and bones. Strain and reserve the cooking stock; you’ll need some for the peas but the rest is great as a base for soups and stews.
To make the pease pudding, gently fry the diced onion in saucepan with a little oil for 10 minutes until softened. Rinse the soaked peas under clean water and add them to the pan with 250ml ham stock and 500ml water. Bring to a simmer and cook gently for about 1½ hours until completely soft and mushy. Transfer to a food processor and blend until smooth (it will thicken more when cooled). Taste and tweak the seasoning with salt, pepper and a dash or two of vinegar to your liking. Leave to cool and then refrigerate.
When you are ready to cook your breakfast, bring a large saucepan of water up to a simmer and add a pinch of salt and 3 tablespoons of vinegar – this is to poach the eggs in. Turn your grill on for the muffins.
Heat 2 tablespoons of oil in a large frying pan and add about 250g of the shredded ham hock. Fry it on a medium heat until the ham is warmed through and starting to colour and crisp slightly on the edges. Keep to one side while you poach your eggs.
Drop the pan of water down to the gentlest of simmers and very gently crack in your eggs. Poach for 3-4 minutes until the whites are set but the yolks are still runny. Depending on your pan you may want to do this in batches. While your eggs poach, toast and butter your muffins.
To serve, top each muffin half with a spoonful of the cold pea pudding, followed by some crispy ham hock and then a poached egg. Finish with a pinch of salt and a few turns of pepper. Eat immediately.