Heat a splash of oil in a large, non-stick frying pan. Fry the bacon until crispy, then put on a plate covered with foil to keep warm.
Using a cookie cutter, cut a hole in 1 slice of bread, then spread mayonnaise on one side of both slices. Fry the bread in the same pan. When browned on one side, flip both over and crack the egg into the hole. Fry for 2-3 mins, then turn down the heat and cover the pan until the white of the egg is set but the yolk is still runny. Remove everything from the pan.
Spread the non-egg slice with the sauce, add the bacon, then top with the egg slice. Halve and tuck in.