Use your thumbs to make the centre hole of the bagel as big as an egg yolk. Squeeze the sausage out of its skin and press the meat onto one side of the bagel, leaving a gap in line with the hole. Place the bagel sausage-side down in a large non-stick frying pan on a medium-low heat and cook for 1 minute. Crack the egg into the hole, season with a pinch of black pepper, then add a splash of water to the pan and cover with a lid. Cook for 8 minutes, or until the sausage has crisped up. Remove the lid and coarsely grate over half the cheese, then use a fish slice to confidently flip the bagel. Grate the remaining cheese onto the sausage, pop the lid back on the pan and cook covered on a low heat for 4 more minutes, or until the cheese has melted. Serve with tomato ketchup and/or brown sauce.