For the Korean pulled pork: Remove the pork from the refrigerator 30 minutes before cooking.
Whisk together the soy sauce, gochujang, hoisin, ketchup, vinegar, honey, sesame oil, five-spice powder and pepper until smooth. Rub the pork with the marinade, cover and refrigerate for at least 4 hours and up to 24 hours.
Prepare the Caja China according to manufacturer's directions. Cook until the pork reaches 190 degrees F, 3 to 3 1/2 hours. Remove the pork, place in a baking pan, tent with foil and let the pork rest for 20 minutes. Reserve the juices for the BBQ sauce.
For the Korean BBQ sauce: Heat the oil in a medium high-sided saute pan. Add the shallots and cook until soft. Whisk in the reserved juices from the pork, the chicken stock, ketchup, gochujang, honey, soy sauce and five-spice powder and cook until slightly reduced. Remove from the heat, add the vinegar and sesame oil.
For the Asian slaw: Whisk together the vinegar, honey, sesame oil, and soy sauce until the honey is dissolved. Add the peas, carrots, cabbage and sesame seeds. Toss to combine; season with salt and pepper. Cover and refrigerate until cold, at least 1 hour.
For serving: Using tongs, pull the pork into bite-size bits. Toss with the BBQ sauce. Mound some slaw on the bottoms of the buns. Top with the pork and the top bun.