Preheat the oven to 350℉. Spray a donut pan with nonstick spray.
PREPARE THE DONUTS:
In a medium bowl, whisk together the flour, sugar, baking powder, cinnamon, salt and nutmeg. In a separate bowl, whisk together the eggs, milk, butter and vanilla. Stir the wet mixture into the dry ingredients- just until incorporated. Stir in the pumpkin.
Spoon the batter into the donut pan so that the cups are nearly full. Bake until the donuts are golden brown, about 15 minutes. Remove from the pan immediately and cool on racks until completely cooled before adding the glaze. Repeat with remaining batter and pan.
PREPARE THE GLAZE:
Whisk the glaze ingredients together, heat a little bit in the microwave until warm. Dunk the tops of the donuts, return to the cooling rack and let the glaze set.