Prepare the Cake: Preheat oven to 350 degrees F. Spray a 9-inch* round cake pan with nonstick cooking spray. (If you don't own a 9-inch cake pan, no problem. Use an 8-inch round pan. If your pan is 2-inch deep, you'll be fine. If it's only 1 1/2 inches deep, fill the pan 2/3 full. Use the remaining batter for a cupcake or two.)
Place cake mix in a medium mixing bowl. Add pumpkin pie spice. Whisk to combine. Add the pumpkin, oil, eggs, and vanilla extract. Whisk until smooth. Batter will be very thick. (Feel free to use a handheld mixer.) Spread batter into prepared pan.
Prepare the Icing: In a medium bowl, whisk cream cheese until smooth. In a small bowl, combine powdered sugar, milk, and vanilla extract. Stir with a spoon until smooth. Add powdered sugar mixture to the cream cheese. Beat until smooth. Spread icing over cooled cake. (If the cake is warm, the icing will melt.)