In a small sauce over medium heat, add butter and bring to a boil. Continue heating over medium heat, constantly stirring until the butter has turned a nutty brown color. It may be hard to see the color if your butter is foaming a lot, so periodically take it off the heat to check the color. Remove from heat, place into a heat proof glass and then into the freezer to cool off for 15-30 minutes. When ready, the butter should still be liquid and just barely warm to the touch.
Preheat oven to 350°F, and prepare 4 donut pans with cooking spray. Next, in a medium mixing bowl, add flour, baking powder, salt, and all the spices. Whisk together to combine, then set aside.
In a large mixing bowl, add oil, cooled brown butter, white sugar, and brown sugar. Beat with a hand-mixer until smooth. Add pumpkin puree and vanilla, then beat again until well-combined and smooth. Finally, add eggs in, one at a time, and beat until smooth.
Add dry ingredients to the wet ingredients, and beat just until smooth.
Add batter to donut pans by spoon or using a piping bag, then transfer to oven to bake for 13 minutes, or until an inserted toothpick comes out clean. Allow to cool completely before coating with cinnamon sugar.