Preheat oven to 375 degrees F (190 degrees C). Lightly grease 2 baking sheets.
Stir flour, brown sugar, 2 teaspoons pumpkin pie spice, baking powder, salt, and baking soda together in a large bowl.
Mix pumpkin, eggs, milk, and butter into flour mixture; beat until completely incorporated.
Spoon pumpkin mixture into a pastry bag fitted with a large star tip with a 1/2-inch opening. Pipe 3-inch circles onto prepared baking sheets.
Bake in the preheated oven until golden brown, about 13 minutes. Remove doughnuts to a wire rack to cool.
Stir confectioner’s sugar, 1/2 teaspoon pumpkin pie spice, and vanilla together in a bowl. Gradually stir in enough milk to reach a glaze consistency. Brush glaze over doughnuts with a pastry brush or spoon over the tops.
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