FOR DOUGHNUTS: In a medium mixing bowl stir together flour, brown sugar, baking powder, pumpkin pie spice, salt, and baking soda. Add pumpkin, eggs, milk, and shortening, butter, or margarine; beat with an electric mixer on low speed till mixed.
Spoon the mixture into a pastry bag fitted with a large star tip with a 1/2 inch opening. Pipe onto 2 greased baking sheets in 3-inch circles. Bake in a preheated 375* oven for 10 minutes, or till golden brown. Cool doughnuts on a rack over waxed paper.
FOR ICING: In a small mixing bowl stir together powdered sugar and vanilla. Stir in enough milk to make a smooth icing of glazing consistency. Spoon icing over doughnuts.