Preheat oven to 325°F. Grease a 9 x 13-inch baking pan with nonstick cooking spray; set aside.
In a large bowl, combine the cake mix, eggs, vanilla, oil, and sour cream. Beat with an electric mixer for 3 minutes on medium speed.
Pour half of the batter into the prepared pan.
In a small bowl, stir together the brown sugar and cinnamon. Sprinkle evenly over the batter in the pan, spreading toward the edges.
Dollop the remaining batter all over the top, using a spatula or the back of a spoon to spread it into an even layer.
Bake for 44 to 48 minutes, until golden brown and a toothpick inserted in the center comes out clean.
Let the cake cool for 5 to 10 minutes, then use a fork to prick holes all over the warm cake.
While the cake is still warm, whisk the powdered sugar, vanilla, and milk together until smooth. It should be thin enough to pour. Add more powdered sugar if too thin, or more milk if too thick.
Pour evenly over the warm cake, then slice and serve warm or at room temperature.