Discard dark green and damaged outer leaves off the head of cabbage and wash it. Cut the cabbage in half, lengthwise, through the core, and then cut each half through the core again. Cut out the core off each cabbage quarters. Wash and peel the carrots.
Slice each quarter of cabbage into thin slices and shred the carrots.
Combine all remaining ingredients for the coleslaw dressing in a bowl. Add sliced cabbage and shredded carrots to the mixing bowl and mix thoroughly with the dressing.
Refrigerate in a food storage container with a lid until ready to use.
Slow Cooker Pulled Pork:*
Combine all the seasoning ingredients for the pork in a bowl and mix well. Place pork shoulder on a sheet of parchment paper and season it generously on all sides.
Preheat a large skillet over medium to medium-high heat and add oil. Sear pork on all sides until it gets a nice sear and then place it into the slow cooker.
Combine apple juice, apple sauce, and chicken stock and pour it all around the pork.
Cover with a lid and cook on LOW for 8-10 hours. (Until internal temperature is between (190° and 203°).
Take the pork out and let it cook a little before pulling. Pull pork apart once it's cooled enough to be handled. As you're working through it, discard excess fat and membrane.
Save enough of the liquid from cooking pork to make the BBQ sauce and discard the rest.
Combine all the ingredients for the sauce in a small sauce pot and bring it to simmer over medium heat.
Let is simmer for about 10 minutes and make sure to sir often.
You can stir all of the sauce with pulled or stir half into pork and serve additional sauce on the side.
If you like extra BBQ sauce, spread some on the bottom bun and then, top it off with 1/3-1/2 cup of pulled pork and about 1/4 cup of coleslaw.
You can add some more toppings to the sandwiches if you wish.***
*If you want to make pulled pork a lot faster, use Instant Pot. It should be ready in about an hour and a half. Use "sear" setting of the Instant Pot to sear the meat first, then add the liquid mixture, close the lid property, and let it cook on high pressure for 70-75 minutes. Let it do a natural release for 10 minutes and then do a manual release. Note: for bone-in pork, add another 10 minutes of cook time. ** Use standard hamburger buns for full size sandwiches or, you can also use slider buns for smaller, bite-sized sandwiches. You should be able to get double the amount of sliders out of this recipe. ***Play around with some additional topping or toppings instead of slaw like Horsey sauce, pickled onions, pickled jalapenos, caramelized onions, raw onions, avocados, and more.