Preheat oven to 325 degrees Fahrenheit (163 degrees Celsius). Adjust the oven rack to the middle-lower position (so that your pot fits).
Cut the fat cap off the large chunk of pork. Slice the pork into about 8 large chunks. You'll want the pork to remain marbled with fat, but if there are giant spots, it's okay to remove them.
In a small bowl, combine the salt, pepper, paprika, cayenne, garlic powder, cumin, and brown sugar. Rub the mixture over the pieces of pork.
In a large Dutch oven over medium-high heat, add the vegetable oil. When the oil is hot, add the pork and sear. When the meat is crispy and brown, flip and continue to sear the other side (I let mine go for about 4 minutes on each side).
Meanwhile, whisk together the mustard, ¾ cup water, apple juice, and apple cider vinegar. Pour over the braised pork, and bring to a simmer. When the mixture is rapidly bubbling, cover and transfer to the preheated oven and bake (braise) for 3-3½ hours or until the meat is fork-tender and the liquid is almost entirely gone.
Remove from the oven and take the lid off. Allow to cool for 10 minutes before transferring the meat to a cutting board or plate. Using two forks, shred them into bite-size pieces. Toss with remaining ¼ cup of water, and serve immediately.
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Notes
*My favorite salts are Morton's Kosher Salt or Iodized Sea Salt. They're much larger than table salt and are both ideal for baking and cooking. I highly recommend using either of them!-If you don't own a Dutch oven, simply braise the meat in a large skillet, pour in the mustard mixture, bring to a simmer, and then transfer everything to a 9x13 ceramic or glass dish. Cover with foil and bake according to the directions.-If you want the meat to taste more like BBQ with a smokier flavor, try adding a few drops of liquid smoke to the liquid mixture.-After adding the water at the very end, you can toss the pork in your favorite barbecue sauce (about ¼ cup) for more flavorful pork. Or, simply top it with sauce!STORE the leftover pulled pork in an airtight container in the refrigerator, where it will stay good for up to 3-4 days. To FREEZE, store the pulled pork for up to 3 months in freezer-safe bags or containers.To REHEAT, thaw in the refrigerator if frozen, then warm in a saucepan over low heat with a splash of apple juice or water to keep it moist. Alternatively, you can reheat it in the microwave, covered, stirring occasionally to ensure even heating.