Season pork with salt, 1 teaspoon ginger, and allspice. Cut 3 cloves of garlic into halves. Make small slits into various areas of the pork and stuff with the garlic. Place into a slow cooker and cook on high for 3 hours. Once finished, shred pork with two forks. Return to slow cooker and keep covered on low until ready to serve.
To prepare the sauce, add 2 teaspoons oil, 2 minced garlic cloves, 1 teaspoon ginger, diced onion, soy sauce, ketchup, honey, rice wine vinegar, Hoisin sauce, and sesame oil to a small saucepan. Whisk together and heat over low heat for 5-8 minutes until thickened. Remove from heat and cover.
Shred cabbage and place into a mixing bowl along with chopped cilantro. Season with cider vinegar and remaining oil.
When ready to serve, place shredded pork onto each roll, drizzle with sauce, and then top with cabbage.