Preheat oven to 375 degrees. Spray a donut pan with non-stick spray.
Add the dry ingredients to the bowl of a stand mixer fitted with the whisk. Mix to combine.
Add the pumpkin, egg, milk, and oil to the bowl and beat until well combined.
Spoon mixture into the donut pan and bake for 8-10 minutes or until a tester comes out clean.
While the donuts are cooling in the pan, melt the butter in a small saucepan over low heat. Whisk constantly and continue cooking until golden, about 8 minutes.
Pour the butter into a small bowl and whisk in the powdered sugar, milk, and vanilla until smooth.
Dip the donut tops (slightly warm or cool) in the glaze and enjoy.