Lay the bread slices flat on a work surface. Spread mayonnaise on one slice and arrange the cheese on the other. Arrange the apple and red onion slices on top of the cheese, then top with arugula and sandwich the bread slices together.
Spread half of the butter on the upward-facing side of the sandwich, then flip that side down into the pan. Butter the other side.
Cook, flipping once, until the bread is golden and the cheese is completely melted. If the sandwich is getting dark too quickly, reduce the heat.
Notes
Mayo + butter combo: Mayonnaise on the inside and butter on the outside gives an extra-crisp, golden crust.Aged cheddar: A sharp aged cheddar stands up to the sweet apple and peppery arugula.Slice thin: Thin apple and onion soften into the sandwich and let the cheese ooze.Low and steady: Cook over medium so the cheese fully melts before the bread over-browns.