Preheat the oven to 350 degrees. Line a baking sheet with parchment paper to prevent sticking.
In a medium bowl, combine the brownie mix, egg, and vegetable oil. Stir until the batter is smooth and fully mixed.
Lightly flour a large piece of parchment paper. Unroll the cinnamon roll dough and lay it flat on the parchment, keeping the dough in one even layer.
Spoon the brownie batter over the cinnamon roll dough. Use a spatula or spoon to carefully spread the batter into an even layer.
Starting from one long edge, gently roll the dough up around the brownie batter to form a log. Roll slowly to keep the swirl neat and even.
Use a sharp knife to slice the rolled dough along the cinnamon roll perforations, creating individual rolls.
Place the sliced rolls onto the prepared baking sheet, leaving a little space between each one. Bake for 15 to 20 minutes, or until the cinnamon roll dough is cooked through and the brownie swirl is set.
Remove the rolls from the oven and immediately spread the included cinnamon roll icing over the tops while the rolls are still warm so it melts smoothly.
Place the chocolate chips in a microwave-safe bowl. Heat in 30-second intervals, stirring between each interval, until fully melted. Drizzle the melted chocolate over the iced rolls just before serving.