Preheat oven to 375°F. Spray two donut pans with nonstick spray.
PREPARE THE DOUGHNUTS:
In a medium mixing bowl, whisk together the flour, brown sugar, baking powder, pumpkin pie spice, salt and baking soda. Add the pumpkin, eggs, milk, and butter; beat with an electric mixer on low speed until combined.
Spoon the mixture into a pastry bag fitted with a large tip with a 1/2-inch opening. Pipe onto donut pans. Bake 10 minutes, or until golden brown. Cool the doughnuts on a rack over waxed paper.
PREPARE THE ICING:
In a small mixing bowl stir together the powdered sugar and vanilla. Stir in enough milk to make a smooth icing of glazing consistency. Spoon the icing over the doughnuts.