Preheat the oven to 375 degrees F (190 degrees C). Line a large rimmed baking sheet with parchment paper and set aside.
Mix together pork and barbecue sauce in a medium bowl until combined; set aside.
Unroll 1 can of the dough onto prepared baking sheet. Sprinkle 3 ounces each of the Cheddar and mozzarella (about 1 1/2 cups total) over dough, leaving a 1/2-inch border around the edges. Spread pork mixture evenly over the top of the cheese, then sprinkle remaining 3 ounces each Cheddar and mozzarella (about 1 1/2 cups total) evenly over pork.
Unroll remaining 1 can dough and place on top of the fillings, stretching as needed. Press edges together, using your fingertips to tightly seal dough.
Whisk together Dijon, garlic powder, and 2 tablespoons of the butter in a small bowl; brush evenly over dough.
Bake in preheated oven until golden brown, about 20 minutes.
Meanwhile, whisk together remaining 2 tablespoons butter and scallions in a small bowl. Remove grilled cheese from oven and brush immediately with scallion butter. Slice and serve with additional barbecue sauce. Recipe developed by Isabelle Easter
