Preheat the oven to 350°F (175°C). Grease a 9x13-inch baking dish.
In a large bowl, combine the cake mix, eggs, sour cream, and vegetable oil. Mix until smooth.
In a separate bowl, stir together the brown sugar and cinnamon.
Pour half of the cake batter into the prepared baking dish.
Sprinkle half of the cinnamon-sugar mixture evenly over the batter.
Spread the remaining batter over the cinnamon layer.
Top with the remaining cinnamon-sugar mixture.
Use a butter knife to gently swirl the cinnamon mixture throughout the batter.
Bake for 35–40 minutes, or until a toothpick inserted in the center comes out clean.
Let the cake cool for 10–15 minutes.
In a small bowl, whisk together the powdered sugar, milk, and vanilla extract until smooth.
Pour the vanilla icing over the warm cake and spread evenly.
Allow the glaze to set before slicing and serving.