Preheat oven to 325 degrees F (165 degrees C). Line a baking sheet with parchment paper.
Stir all-purpose flour, brown sugar, baking powder, pumpkin pie spice, salt, and baking soda together in a bowl; add pumpkin puree, eggs, milk, and softened butter. Beat mixture with an electric hand mixer on low speed until combined into a batter; spoon into a donut pan.
Bake in preheated oven until donuts spring back when touched, 8 to 10 minutes.
Beat confectioners' sugar, melted butter, water, and vanilla extract together in a bowl until smooth. Dip warm donuts into the glaze and let any excess glaze drip back into bowl before placing glazed donut on prepared baking sheet to cool.
