Bake the vanilla cake in a 9x13 inch pan according to package instructions. Let it cool for 10 minutes.
While the cake is still warm, use the handle of a wooden spoon to poke holes all over the top.
In a bowl, whisk together 2 cups of milk and the pudding mix until smooth. Pour over the cake, filling the holes.
In a small bowl, mix brown sugar and cinnamon. Sprinkle evenly over the cake.
In another bowl, whisk powdered sugar, 2 tablespoons milk, and vanilla extract until smooth. Drizzle over the cake.
Chill for at least an hour or serve slightly warm—your call!