Place the bread in a toaster set to your desired level of darkness. Heat a skillet over medium-high heat. Fry the hamburger patty for about 5 minutes per side, until cooked through. While the patty is cooking, spread mayonnaise on one side of each piece of toasted bread and place them on a plate, mayonnaise side up.
Once the patty is cooked through, transfer it to one slice of bread. Crack the egg into the skillet and fry until the yolk is firm, about 3 minutes if you cover the pan. Place the egg onto the patty when done and top with a slice of cheese. Top with the other slice of bread. When finished, it should look like a sandwich.