Melt the butter in a small saucepan over medium-low heat. Add the onion, season with salt and pepper, and cook until brown and caramelized, stirring occasionally, about 30 minutes.
Mash together the potatoes, caramelized onions, cheese, softened butter, salt, and pepper.
Cut the sausages into 2-inch pieces and encase the sausage pieces in the potato mixture.
Add the flour, eggs, and bread crumbs to 3 separate shallow bowls.
Coat the sausages first in the flour, then the eggs, followed by the bread crumbs, then once more in the eggs and bread crumbs. (In order to reduce mess, maintain one "dry" hand for coating in the flour and bread crumbs and one "wet" for coating in the egg.
Heat the oil in a large pot until it reaches 325˚F (160°C). If you don't have a thermometer, you can test the oil by dropping in a small piece of white bread. If it sizzles and browns within 45 seconds, the oil is ready.
Carefully deep-fry the bangers and mash bombs 4 at a time for a few minutes, until golden brown.
Drain on a paper towel, sprinkle with salt, and serve with warm gravy.