Preheat an oven to 325 degrees F (165 degrees C). Grease a 9x13-inch cake pan.
In a bowl, whisk together the cake mix, buttermilk, eggs, baking soda, and vegetable oil until thoroughly combined. In a separate bowl, stir the brown sugar and cinnamon together. Pour batter into the prepared cake pan, and spoon the cinnamon sugar onto the batter in several places. Using the spoon, gently swirl the cinnamon sugar into the cake batter, leaving streaks and ribbons of sugar mixture (do not overmix).
Bake cake in the preheated oven until a toothpick inserted into the center of the cake comes out clean, 30 to 35 minutes. Remove cake from oven, and punch holes all over the cake with a table knife.
Whisk milk, vanilla extract, and confectioners' sugar together in a bowl to form a smooth, runny icing; pour the icing all over the cake, allowing it to soak into the holes. Allow cake to cool before serving.
