Place two slices of the bread on a flat working surface and top each with a slice of cheese. Top each slice of cheese with half of the sliced peach. Layer the arugula over the peaches on each sandwich. Add the second slice of cheese and then top each with another slice of bread.

Melt the butter in a large skillet over medium heat. Add the sandwiches to the skillet and cook for 2 to 3 minutes, until golden brown. Flip and cook for another 2 to 3 minutes, until the cheese is melted. Serve warm.