Preheat the oven to 350° F. Spray donut molds with nonstick spray and set aside. In a large bowl, whisk together the flour, baking powder, baking soda, salt, pumpkin pie spice, and brown sugar.
In a large measuring cup or medium bowl, whisk together the pumpkin puree, melted butter, egg, milk, and vanilla. Pour this mixture into the dry ingredients, and mix until just combined.
Scoop this mixture into a piping/ziploc bag to pipe into the donut pan. You could also just spoon the batter into the pan if you prefer. Fill the donut pan ½ full with batter. Don't overfill the donuts because they rise while baking. If you are using silicone donut molds, place them on a baking sheet before putting them into the oven.
Bake for 9-12 minutes until golden brown on top. Transfer the donut pan to a wire rack and allow the donuts to cool for about 5 minutes. Carefully take the donuts out of the pan and allow them to cool for 10-15 minutes before coating them with sugar.
In a shallow bowl, mix the sugar and pumpkin pie spice. Brush the top, bottom, and sides of the donuts with a little bit of melted butter, and then coat with the sugar mixture. Shake off any excess and repeat for the rest.