Preheat the oven to 325 degrees
Combine cake mix, sour cream, eggs, and oil in a bowl. Beat with a hand mixer until smooth.
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 until a toothpick inserted into the center comes out clean, about 45 minutes.
Meanwhile, mix confectioners' sugar, milk, and vanilla together in a bowl.
When cake is done, remove from oven. 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.