Heat the oil in a small frying pan over medium heat.
Shape the breakfast sausage into a round patty, slightly larger than an English muffin. Using a small glass cup, stamp out a hole in the middle of the patty.
Place the patty into the pan and cook until brown and crispy, about 2 minutes. Flip the patty and crack the egg into the hole. Cover the pan with the lid and cook for 1 minute. Place the cheese on top of the patty. Cover the pan with the lid and cook until the cheese melts, about 30 seconds. Sprinkle a pinch of salt and pepper on the cheese. Serve the patty on the English muffin.
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