Preheat the oven to 325 degrees F (165 degrees C). Lightly grease 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 the batter into the prepared baking pan.
Combine brown sugar and cinnamon together in a small bowl; sprinkle over batter in the pan. Carefully pour remainder of batter over brown sugar-cinnamon layer.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, 45 to 50 minutes.
Combine confectioners' sugar, milk, and vanilla together in a bowl. Poke holes in hot cake with a fork; pour confectioners' sugar mixture over top of cake.
Let cake cool completely before serving, about 30 minutes.
