Preheat the oven to 350°F and lightly grease two 6-well donut pans with nonstick baking spray.
In a large bowl, whisk together the milk, pumpkin puree, melted butter, egg, and vanilla.
Add the flour, brown sugar, baking powder, and pumpkin pie spice. Mix just until no large lumps of flour remain, since you don't want to overmix.
Transfer the batter to a piping bag, or a Ziploc bag with the corner cut off, and pipe it into the prepared pans. Fill each section about ½ full, since these donuts rise quite a bit.
Bake for 16 to 18 minutes, or until a toothpick inserted into a donut comes out clean.
Move the donuts to a wire rack and cool completely before icing.
For the icing, whisk together the powdered sugar, vanilla, and 2 tablespoons of milk. Add another tablespoon of milk for a thinner icing, or more powdered sugar for a thicker one. It can be thick and spreadable or thin like a glaze, whatever you prefer.
Ice the donuts and sprinkle with a pinch of pumpkin pie spice. Let the icing set for about 10 minutes, then serve.
Store leftover donuts in an airtight container in the fridge for up to 3 days.