Preheat the oven to 350ºF and spray a donut pan with non-stick cooking spray. Set it aside.
In a medium bowl, whisk together the pumpkin puree, brown sugar, vegetable oil, and milk.
In a smaller bowl, combine the flour, baking powder, pumpkin pie spice, cinnamon, nutmeg, and salt. Gradually add the dry ingredients to the wet ingredients, mixing with a spatula until just combined.
Transfer the batter to a piping bag or a ziplock bag with the corner cut off. Pipe the batter into each donut cavity, dividing it evenly among the six donuts.
Bake for 15-16 minutes, or until a toothpick inserted into the center comes out clean. Allow the donuts to cool in the pan for 10 minutes.
Flip the pan upside down to release the donuts onto a cooling rack and let them cool for at least 10 more minutes.
For the cinnamon sugar coating, melt 2 tablespoons of butter in the microwave. Use a pastry brush to coat the cooled donuts with melted butter, or quickly dip each side of the donut in the butter if you don’t have a brush.
Combine the cinnamon sugar mixture in a bowl. Place each donut in the sugar mixture, flipping to coat both sides.