For the lemon and tarragon pulled pork, in a saucepan heat 1 litre/1¾ pints water, the salt, Demerara sugar, bay leaves, thyme, star anise and the lemon rind. Boil until the salt and sugar have dissolved then remove from the heat and leave to cool.
When the brine is cold, place the pork shoulder into a container and pour over the chilled brine. Cover and place into the fridge for 24 hours.
When ready to cook the pork, preheat the oven to 140C/275F/Gas 1.
Remove the pork from the marinade, pat dry and place onto a sheet of aluminium foil. Drizzle with the oil and wrap the meat in the foil, sealing the edges to form a parcel. Place in a roasting tin and transfer to the oven to slow cook for at least four hours, or until the pork is very well done and tender.
Remove from the oven and leave to cool, still wrapped, for 25-30 minutes.
Meanwhile, in a large pan, combine the white wine vinegar, chicken stock, brown sugar, white wine, grated garlic, honey and bay leaves and bring to the boil. Turn the heat down to a low simmer.
Open the wrapped pork and, using two forks, pull the pork apart, shredding it all up. Place the pork mix into the gently simmering stock and cook for 35-40 minutes, letting the stock reduce to a sticky glaze.
Meanwhile, for the walnut mayonnaise, place the egg yolks, mustard and white wine vinegar into a food processor and blend until smooth. Slowly add the vegetable oil in a thin, steady stream while the motor is still running.
When all the vegetable oil is combined, add the walnut oil with the motor running. Blend until the mayonnaise has emulsified then season with salt, cayenne pepper and lemon juice. Stir in the walnuts.
For the coleslaw, mix together the carrot, cabbage and onion with a little mayonnaise.
When ready to serve, season the pork with the lemon juice, tarragon, salt and pepper.
To serve, tear apart the pitta bread apart, put in a big blob of coleslaw, add the pulled pork and lettuce (if using).