Preheat oven to 350 degrees F (175 degrees C). Grease a 9x13-inch cake pan.
In a mixing bowl, whisk together the cake mix, eggs, sour cream, vegetable oil, and water to make a smooth batter. In a separate bowl, stir together the brown sugar with cinnamon. Pour half the cake batter into the prepared baking pan, and sprinkle the brown sugar-cinnamon mixture over the batter. Pour the remaining batter over the cinnamon mixture, and gently swirl the batter a few times with a table knife.
Bake the cake in the preheated oven until lightly golden brown and a toothpick inserted into the center of the cake comes out clean, about 40 minutes.
Stir the confectioners' sugar, milk, and vanilla extract together in a bowl to make a smooth, runny icing. Pour the icing over the cake while it's still hot; allow to cool before serving.