Heat the vegetable oil in a large pan over medium heat. Add the onion and sauté until it turns a light golden brown, about 2 minutes. Add the garlic and apple and continue cooking until the apples are soft, about 2 minutes.
Add the cabbage and cook, stirring frequently, until it has wilted, then season with salt and freshly ground pepper. Continue cooking for about 5 minutes, or until the cabbage has caramelized and the mixture is a rich golden brown. Transfer to a bowl and set aside.
While the cabbage is cooking, slice the pork as thinly as possible. Stir together the ketchup, apple cider vinegar and brown sugar until the sugar has dissolved. Add a dash of hot sauce if desired.
After removing the cabbage mixture, add the sauce to the pan and heat to bubbling, about 30 seconds. Add the sliced pork and toss until the individual slices are nicely coated. There won't be a lot of excess sauce.
Place a slice of cheese on both the top and bottom of each roll and toast just until the cheese melts. Place a serving of pork on the bottom of each bun and top with the warm slaw. Serve immediately.
Tips for Making This Recipe
In place of leftover pork, you can roast a small pork tenderloin in a 425°F oven for about 20 minutes to medium doneness. Allow it to rest for at least 5 minutes and slice it as thinly as possible.