Preheat the oven to 350°F (175°C). Grease a standard donut pan with nonstick cooking spray.

In a large bowl, whisk together the flour, baking powder, pumpkin pie spice, and salt.

In a separate bowl, whisk together the brown sugar, pumpkin puree, melted butter, egg, milk, and vanilla extract until smooth.

Pour the wet ingredients into the dry ingredients and stir until just combined. Do not overmix.

Transfer the batter to a piping bag or zip-top bag with the corner cut off. Pipe the batter into the donut cavities, filling each about two-thirds full.

Bake for 12 to 14 minutes, until the donuts spring back when lightly touched and a toothpick inserted comes out clean.

Let the donuts cool in the pan for 2 minutes, then carefully turn them out onto a wire rack.

For the coating, mix the granulated sugar and cinnamon in a shallow bowl. Brush each warm donut all over with melted butter, then roll in the cinnamon sugar mixture until fully coated.

Serve warm or at room temperature.