For the beef, preheat the oven 130C/110C Fan/Gas ¾.
In a heavy bottomed, ovenproof dish with a tight fitting lid, add the beef shin, oxtail, vegetables and herbs. Cover with 2 litres/3½ pints water and place a circle of parchment paper on top. Cover with the lid and cook in the oven for 5 hours.
When cooked, remove the beef shin, oxtail, carrots and celery from cooking liquor and set aside until cool enough to handle.
Remove the meat from the bones. Discard the bones and put the meat to one side. Set half of the cooking liquor to one side for serving. Heat the other half in the dish, or a pan, on the hob, to reduce it in volume by three-quarters.
For the pudding, preheat the oven to 160C/140C Fan/Gas 3.
Heat the dripping in a frying pan. Fry the bread until golden-brown and crisp on both sides, but not fried all the way through. Brush liberally with mustard.
Break the cooked beef shin and oxtail into rough flakes. Roughly chop the cooked carrots and celery.
Make a layer of fried bread in the bottom of the baking dish. Sprinkle with a third of the cheese, half of the cooked meat and half of the vegetables. Season with salt and pepper. Make another layer of bread and cheese, then another layer of meat and vegetables. Finish with a final layer of bread and cheese.
Gently pour over the reduced beef stock then bake for 20 minutes. Remove from the oven and allow to stand for 5 minutes before serving.
For the broccoli, heat the oil in a deep-fat fryer to 180C, or half-fill a deep pan with oil and heat until a breadcrumb sizzles and turns brown when dropped in. (Caution: Hot oil can be dangerous. Do not leave unattended.)
Cook the broccoli in salted boiling water until tender but still with a bite. Refresh in ice-cold water and drain.
Mix the mayonnaise and horseradish together. Add a little hot water to give it a pouring consistency.
Dip the broccoli in the milk, shaking off any excess, then coat evenly in flour. Repeat the process once more and then carefully lower into the deep-fat fryer. Cook for a few minutes until golden-brown. Drain on kitchen paper and season with a little salt. Place on a serving dish and dress with the horseradish and mayonnaise mixture.
Serve the broccoli immediately with the hot pudding.