Preheat oven to 350 degrees. Prepare donut pans by greasing with nonstick cooking spray.
In a large bowl, whisk together the gluten free baking flour, baking powder, salt, baking soda, pumpkin spice, and cinnamon. Set aside.
In a separate bowl, combine the pumpkin puree, brown sugar, canola oil, melted butter, vanilla extract, and egg. Whisk to mix.
Add the wet ingredients to the dry ingredients. Stir until just combined.
Spoon the mixture into a piping bag and fill each donut cavity with batter.
Bake for 20 minutes or until donut springs back when pressed with a finger.
Allow donuts to cool in pan for 5 minutes, then remove to cooling rack to cool completely.
While the donuts are cooling, make the glaze by combining the powdered sugar, pumpkin pie spice, vanilla extract, and milk in a medium bowl. Whisk until smooth.
Dip each cooled donut in the glaze, then place back on wire rack until the glaze has set.