Preheat oven to 350°F (175°C) and grease a donut pan.
Brown the butter in a saucepan over medium heat until golden and nutty. Let cool slightly.
In a large bowl, whisk together the browned butter, pumpkin purée, brown sugar, egg, vanilla, and milk until smooth.
In another bowl, whisk together the flour, baking powder, baking soda, pumpkin pie spice, cinnamon, and salt.
Add dry ingredients to wet and mix until just combined.
Spoon or pipe the batter into the donut pan, filling each about ¾ full.
Bake for 10–12 minutes, or until a toothpick comes out clean.
Let cool for 5 minutes, then remove from the pan.
Dip donuts in icing.