Preheat oven to 325 degrees F (165 degrees C). Lightly oil a 9x13-inch baking pan.
Combine cake mix, sour cream, eggs, and oil in a bowl. Beat with an electric mixer for 3 minutes. Pour half of the batter into the prepared baking pan.
Mix brown sugar and cinnamon together in a small bowl. Sprinkle over batter in the pan. Carefully pour the remainder of the batter over the cinnamon-sugar layer.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, 45 to 50 minutes.
Mix confectioners' sugar, milk, and vanilla together in a bowl. Poke holes in the hot cake with a fork and pour sugar mixture over the top of the cake. Let cake cool completely before serving, about 30 minutes.