Heat the grill to medium. Divide the sausagemeat evenly into 4 and mould into patties. Grill for 15-20 minutes, turning halfway through. For the last 3-4 minutes, add a cheese slice to each patty and grill until melted.
Meanwhile, heat 2 tbsp of the oil in a large frying pan. Add the onion and fry over a medium heat for 10-15minutes, stirring occasionally – you want it to soften and caramelise. Transfer to a plate and keep warm.
Turn up the heat under the pan, add the remaining oil, then fry the eggs to your liking.
Meanwhile toast the muffins. Lightly butter the bases, then top each with a sausage patty, a fried egg and some onions. Replace the muffin tops, then serve.