Whisk together the ketchup, vinegar, brown sugar, and soy sauce in a small saucepan. Bring to a simmer over medium heat, reduce the heat to low, and let gently bubble for 10 minutes to let the flavors develop. Taste and adjust the seasoning with salt and pepper. Remove from the heat. Continue with the recipe or cool the barbecue sauce, cover, and refrigerate for up to a week; reheat gently over medium-low heat.
Rinse and drain the jackfruit in a colander, pressing down to extract as much water as possible from it. Pat dry. Cut and discard the core and any seeds. Heat the oil in a medium saucepan over medium heat. When it’s hot, add the onion and garlic and cook, stirring a few times, until softened, 4 to 5 minutes. Add the jackfruit, paprika, and salt and pepper to taste, and stir until well mixed. Add the stock, stir to combine, bring to a simmer, and let continue to simmer, stirring occasionally, until the jackfruit has softened and there is no liquid left in the pan, 10 to 15 minutes. Once it’s tender, mash down on it with the back of a spoon or a potato masher to separate the jackfruit into pulled pork–like shreds.
While the jackfruit mixture is simmering, heat the oven to 400°F.
When the jackfruit is ready, add half the barbecue sauce to the pan and stir to coat well. Transfer the jackfruit to a baking sheet and spread into a single layer. Bake until the jackfruit caramelizes and crisps in places, 15 to 20 minutes.
Split the rolls; toast in the oven if you like. Divide the jackfruit among the rolls and serve with the remaining barbecue sauce on the side.