Heat a nonstick or cast iron skillet or griddle over medium heat.
Spread half of each tortilla with mustard. On that half, spread a layer of cheese and sprinkle with fresh ground black pepper. Top with a layer of apples and then the arugula. Fold tortilla over and place in hot pan with the side with cheese down first.
Press gently on quesadilla. When bottom side is browning and cheese is melting, flip over and toast the other side. When that side is golden, remove with a spatula and serve.