Heat the oil in a non-stick pan over medium heat. Cook the sausage patty on both sides to an internal temperature of 150℉ (65℃).
Heat a nonstick pan over medium heat. Melt the butter in the pan; crack the eggs into the pan. Cook until the whites of the eggs are firm. Gently flip over and top with cheese to melt.
Toast the bagel. Arrange the sausage and eggs onto the bagel and cut the sandwich in half.