Hard boil four of the five eggs. Cool under cold water and remove the shells. Remove any skin from the black pudding then chop it and mix with the sausage meat (pork mince) in a bowl and season well with salt and freshly ground black pepper. Divide the meat mixture into four equal portions then on a lightly floured surface, roll each piece into a circle about 5" (12.5cm) in diameter. Season the flour with salt and freshly-ground pepper and coat each of the hard-boiled eggs in it. Place an egg into the centre of each circle of sausage meat and mould the meat evenly around the egg, making sure it fits closely and completely.
Beat the fifth egg and use it to seal the joins in the sausage meat, pinching the edges well together. Mould each Scotch Egg into a good shape and brush them all over with the beaten egg. Cut the crusts off the bread and crumble the remaining white bread between your fingers to make small breadcrumbs (if you have a food processor, you can use the chopper attachment for this). Toss the Scotch Eggs into the breadcrumbs to coat them evenly and completely. Press the breadcrumbs in firmly.
Since the sausage meat is raw, it's important that you don't try to hurry the frying, so fry the eggs over a medium heat in the deep fat until golden brown, then remove with a slotted spoon and drain on soft, absorbent kitchen paper. When cool, cut in half lengthways and serve.
If you don't have a deep fryer, you can cook Black Pudding Scotch Eggs in a frying pan with 1" (2.5cm) of oil but you must keep turning them frequently to ensure it is cooked and browned on all sides.