Preheat oven to 350 degrees F. Mist 8 wells of a donut pan with nonstick cooking spray and set aside.
In a medium bowl, whisk to combine the flour, baking powder, pumpkin pie spice, nutmeg, and salt; set aside. In a small bowl, whisk to combine the pumpkin puree, brown sugar, egg, milk, and coconut oil. Pour the wet ingredients into the dry and mix until just combined.
Scoop the batter into the donut wells, spreading to coat the bottom and filling each well approximately half-way. Bake for 12-14 minutes or until set. Remove from oven and remove to a cooling rack to cool completely.
Meanwhile, mix together the ingredients for the glaze in a shallow bowl. Dip cooled donuts into the glaze, coating the top side of each. Alternately, use a knife to spread a thin layer of glaze over the top of each donut. Serve immediately.