Preheat the oven to 350°F (180°C). Grease a 12-cavity (or two six-cavity) nonstick donut pan with cooking spray.
In a bowl, add the flour, baking powder, baking soda, pumpkin pie spice and sea salt and whisk to combine.
Place the oil, sugar and vanilla extract in a large bowl and whisk to combine. Whisk in the eggs until the mixture is smooth. Add the pumpkin puree and whisk until fully combined.
Add the dry ingredients to the wet mixture and fold together until just combined (do not overmix).
Transfer the batter to a piping bag fitted with a large plain tip, or to a resealable plastic bag with a ⅜-inch (1 cm) hole snipped in one corner, and pipe the batter into each cavity in the pan until they are three-quarters full (or simply spoon the batter in). Bake for 15–17 minutes—when they are ready, the tops of the doughnuts will spring back when lightly touched. Let the doughnuts cool in the pan for 5 minutes, then turn them out onto a wire rack.
Meanwhile, to make the pumpkin pie spice sugar, place the sugar and spice in a shallow bowl and mix well.
While the donuts are still warm, brush both sides with the melted butter and toss them in the pumpkin pie spice sugar. Return to the racks. Donuts are best eaten the day they are made, but can be stored in an airtight container at room temperature for up to 1 day.