Whisk together salt and white sugar in a large bowl.
Rinse pork in cold water and trim any excess fat.
Place shoulder in a raised edge pan. Coat all sides heavily with salt and sugar mix. Cover in foil and refrigerate at least 8 hours, 24 hours is optimal.
Whisk together all ingredients for the aioli, seal in an air tight container in the refrigerator.
Whisk together brown sugar and Korean red pepper.
After brine process is complete, drain any excess liquid and brine from the pan, and insert a reliable leave-in thermometer into the thickest portion of the meat.
Light grill and set up for two zone cooking, establishing temperature at 225°F to 250°F.
Place pork on grill over indirect heat, close lid and allow to cook until internal temperature is 180°F.
Remove pork from grill, baste with juices collected in the pan, and coat top and sides with the brown sugar and gochugara mixture.
Cover with foil, increase grill temp to 285°F and return pork to indirect heat until it reaches an internal temperature of 205°F. About another 90 minutes.
Remove from grill, allow to rest for 30-45 minutes.
Using a fork, pull pork into a shredded consistency and incorporate any juices from the pan.
Place the pork inside bibb lettuce. Top with kimchi and aioli to serve.