Place half the slices of sharp cheddar on one piece of the bread. Top with sliced apples.
In a medium bowl, toss arugula, a drizzle of olive oil, mustard, salt and pepper. Place the arugula mix atop the apple slices and cover with more cheese, and finish with the other slice of bread.
On a preheated pan or grill, add a drizzle of olive oil. Place sandwich and pan fry or grill until the cheese is melted and the bread is toasted on both sides. Enjoy this updated twist on the classic grilled cheese!