Preheat oven to 375°. Spray a rimmed baking sheet with cooking spray.
Season pork with salt and pepper. In a large skillet, heat oil over medium-high heat. Add tenderloins; cook, turning occasionally, until browned, approximately 7 minutes. Place pork on prepared baking sheet.
Bake until a meat thermometer inserted in tenderloin registers 145°, approximately 22 minutes. Let rest for 20 minutes.
Wrap tenderloins in foil. Refrigerate until thoroughly chilled. Cut into ¼-inch slices.
Spread cut sides of kaiser rolls with Caramelized-Onion Mayonnaise and mustard. Fill rolls with pork, apple slices, cheese, and arugula.