1. Prepare the Pulled Pork: Place the pork shoulder in the slow cooker. In a medium bowl, whisk together the chicken stock, honey, balsamic vinegar, hoisin sauce, blackberry jam, garlic, and onion, and pour the sauce evenly over the pork shoulder. Place the lid on the slow cooker and cook on low for 8 to 9 hours, or until the pork registers 165 degrees F on a thermometer, or is no longer pink.
2. Prepare the Asian Slaw: Meanwhile, place the cabbage in a large bowl. In a small bowl, whisk together the lime juice, vegetable oil, soy sauce, sesame oil and sugar, then drizzle over the cabbage. Toss to combine, making sure all of the cabbage is evenly moistened. Gently stir in the green onions. Cover and refrigerate at least 2 hours.
3. Finish the Pulled Pork: When the pork is done cooking, use tongs to remove it from the slow cooker and place on a large plate. Using two forks, shred the pork.
4. Pour the reserved cooking liquid from the slow cooker into a medium saucepan set over medium-high heat. In a small bowl, whisk together the cold water and cornstarch, then slowly whisk the mixture into the reserved cooking liquid. Cook, whisking constantly, until the sauce boils and thickens, about 5 to 10 minutes.
5. Return the pulled pork to the slow cooker and pour the thickened sauce over top, tossing to combine. Keep the slow cooker on low to keep the meat warm as sandwiches are assembled. The pulled pork and Asian slaw should be stored in separate air tight containers in the refrigerator for up to 4 days.