Combine the brown sugar and cinnamon in a small bowl. Sprinkle the mixture evenly over the buttered dough, keeping the 1-inch border uncovered. Scatter the diced apple over the dough, again keeping the 1-inch border uncovered. Roll the dough into a log starting with the edge closest to you. Roll the dough tightly, making sure to roll evenly and push out any air pockets. When you get to the uncovered edge of the dough, press the dough onto the roll to seal it together.