In a medium saucepan, combine the blackberry sauce ingredients and stir gently. Bring the heat up to medium-low and allow the berries to gently cook until they release their juices, breaking the berries apart as they cook, until thickened, around 20-25 minutes. Let cool, then blend or puree in a high-powered blender or food processor until smooth, around 3 minutes. Sauce can be prepared in advance and stored in an air-tight container in the fridge for up to 3 days.
Next, Iin a medium-sized bowl, add the blackberry sauce, balsamic vinegar, ketchup, molasses, dijon, and salt, and whisk to combine.
In the bottom of a 6-qt Instant Pot, add ½-1 cup water as needed for sealing, then layer the chicken in the bottom. Top the chicken with the blackberry BBQ sauce, then pile on the onion, garlic, and jalapeños. Close the lid and set the Instant Pot to high pressure for 15 minutes, then allow to naturally release for 10 minutes before venting and opening the Instant Pot. Use a meat thermometer to check that the internal temperature of the chicken is 165 degrees, then use a hand-mixer or two forks to shred the chicken inside the Instant Pot.
Pile the shredded chicken onto sturdy buns and add your favorite toppings such as bacon and monterey jack cheese, and serve!