Preheat the oven to 350 degrees. Spray two doughnut pans with cooking spray.
In a mixing bowl, combine the cake mix with the pumpkin puree and pumpkin spice. Stir well. Transfer the mixture to either a pastry bag or food storage bag. Snip off the end and squeeze the dough into the pans. Bake for 8-10 minutes or until a toothpick inserted into the center comes out clean.
Let cool for at least 10 minutes in the pan then transfer to a wire rack to finish cooling.
Make the frosting by beating together the cream cheese and butter until smooth. Add the powdered sugar and vanilla and beat well. Add milk a little at a time until you have a thick glaze. Transfer to a plate.
To glaze the doughnuts, dip into the glaze and turn the doughnut a little to ensure the top of the doughnut is completely covered.