Preheat the oven to 350°F (177°C). Spray a 6-count donut pan with nonstick cooking spray and set aside.
In a large mixing bowl whisk together the flour, baking powder, pumpkin pie spice, nutmeg, and salt.
In a separate mixing bowl, mix together the pumpkin, brown sugar, milk, egg, butter, and vanilla extract until fully combined. Add the dry ingredients to the wet ingredients and mix until just combined.
Pipe or spoon the batter evenly between all 6 donut cavities. Bake at 350°F for 10 to 12 minutes or until a toothpick inserted into the donuts comes out clean.
Remove from oven and allow to cool in the pan for 5 minutes, then remove the donuts from the pan and transfer to a wire rack to cool.