Preheat oven to 350 degrees F. Grease doughnut pans.
In a mixing bowl, mix flour, baking powder, salt and pumpkin spice together. Set aside.
In another bowl, whisk the sugars together and then add the oil, eggs, pumpkin puree, and vanilla. Add this liquid mixture to the dry ingredients mixing until just combined. Pipe or spoon batter into donut pans (about 1/4 cup into each mold. You might have to bake multiple batches depending on how many donut pans you have.).
Bake for 13-16 minutes until toothpick comes out clean. Allow donuts to cool in pan for 5 minutes and then transfer to a cooling rack. Make the frosting while donuts are cooling and then dip.
For the Frosting:
In a small pot, heat butter, milk, and corn syrup over medium heat until butter melts. Turn off heat and whisk in the chocolate and vanilla until chocolate melts. Whisk in the powdered sugar until dissolved and smooth. Allow it to cool down and thicken for about 5 minutes before dipping the donuts in. Once dipped, sprinkle each donut with chocolate sprinkles.