Preheat oven to 350°F (175°C) and spray donut molds with nonstick spray.
Mix flour, baking powder, salt, pumpkin pie spice, and brown sugar.
Whisk pumpkin puree, melted butter, egg, milk, and vanilla.
Combine wet and dry mixtures, stirring gently until just mixed.
Fill molds halfway with batter.
Bake 9–12 minutes, cool 5 minutes, then transfer to a rack.
Mix sugar with pumpkin pie spice, brush donuts with butter, and coat in the sugar mix.