Trim any excess fat from the pork shoulder and place it in the slow cooker.
In a medium bowl, whisk together the honey, balsamic vinegar, jam, hoisin, chicken broth, garlic and onion, and then pour the sauce over the pork shoulder. Cover the slow cooker and cook the pork shoulder on LOW for 8 to 9 hours until it is no longer pink and is cooked throughout.
Using tongs, remove the pork shoulder from the slow cooker (reserving the liquids) and place it on a cutting board. Using two forks, shred the pork shoulder into smaller pieces. Place the pulled pork in a large bowl.
In a small bowl, whisk together the cornstarch with 3 tablespoons of cold water to make a slurry.
Transfer the liquids from the slow cooker into a medium saucepan set over medium-high heat and whisk in the slurry. Bring the sauce to a boil and cook it until it reduces and has thickened slightly, about 3 minutes. Pour the sauce over the pulled pork, tossing to combine.
Halve the buns. Divide the pulled pork between the buns, topping the pork with a portion of theAsian Slaw, and serve immediately.
If you don't have blackberry jam, you can use any other variety of jam, such as raspberry or plum.
Depending on your preferences, you may want to pour only a portion of the sauce over the pulled pork. Any leftover sauce can be served with other proteins or veggies.
The pulled pork and slaw can also be served in tortillas for pulled pork tacos.