1) Heat a panini press or an electric griddle to 350°F.
2) Generously butter one side of each slice of bread.
3) Flip four slices over so the butter side is facing down. Top each with the sliced apple, cheddar, arugula, and the remaining slices of bread (butter side up).
4) Place the prepared sandwiches on the hot press or griddle. If using a press, toast the sandwiches until both sides are a nice golden brown and the cheese has melted within. If using a griddle, toast the sandwiches for 5 minutes on each side, then transfer them to a parchment-lined baking sheet, and finish baking in a preheated 350°F oven for another 5 minutes; this allows the cheese to melt in the center of the sandwich.
5) Slice sandwiches on the diagonal, and serve warm.
Yield: 4 sandwiches.