For the sticky pork: Preheat the oven to 300 degrees F.
Heat the olive oil in a Dutch oven over medium-high heat. Season the pork butt with salt and pepper. Add the pork to the pot and brown on all sides, about 5 minutes each side, 15 to 20 minutes total. Remove the pork to a plate and set aside. Leave any rendered fat in the pot and place it back on the stovetop.
Add the garlic and onions to the pot and cook until the onions have softened slightly, about 2 minutes. Add the apple juice, deglazing the bottom of the pot. Scrape all the browned bits from the bottom. Add the applesauce, soy sauce, brown sugar and vinegar and stir together. Carefully return the pork to the pot and cover. Transfer the pot to the oven and braise until the pork is very tender, about 3 hours.
For the apple saw: Meanwhile toss the cabbage, carrots, green onions and apples in a large bowl. In a separate bowl, whisk together the mayonnaise, parsley, mustard, honey, vinegar and salt in a small bowl until well combined. Add the dressing to the apple mixture and toss until well coated.
When the pork is ready, carefully remove the lid and serve straight from the pot. Use a fork and a knife to slice and serve with the braising liquid and the slaw alongside.