Special equipment: three 6-cup nonstick doughnut pans
For the doughnuts: Preheat the oven to 350 degrees F.
Whisk the flour, pumpkin pie spice, baking powder and salt together in a medium bowl.
Beat the pumpkin puree, sugar, vegetable oil, butter, vanilla and eggs and together in another large bowl until smooth.
Add the flour mixture to the pumpkin mixture and beat until just combined.
Fill three 6-cup nonstick doughnut pans with the batter so that each mold is almost full. Smooth the tops with a moistened finger.
Bake the doughnuts until a wooden pick inserted into the center comes out clean, about 15 minutes. Invert the doughnuts onto a cooling rack.
For the topping: Stir the sugar and cinnamon together in a shallow bowl. Working one at a time, brush the doughnuts all over with butter, then dip them in the cinnamon sugar to coat. Serve warm or at room temperature.