In a large bowl, whisk together the flour, baking soda, cinnamon, ginger, cardamom, and salt.
In a small bowl, combine the pumpkin, brown sugar, egg, and oil, and whisk until smooth and blended.
Bake in the center rack of the oven until the top of the donut springs back when lightly pressed, about 13 minutes. Remove from oven and cool on the counter for 3 or 4 minutes. Invert pan and gently release donuts onto a rack to cool for another 10 minutes or so. Lightly dust with confectioners' sugar or make the brown butter glaze below.
Melt butter in a medium saucepan over medium heat until it bubbles and then turns nutty brown, 4 to 5 minutes. Add confectioner's sugar, vanilla, and milk to browned butter and whisk until silky smooth with no lumps. If the glaze solidifies a bit as it cools, add more milk, a few drops at a time as needed.
Dip cooled doughnuts into glaze, allowing any excess frosting to drip back into bowl.
All-purpose flour or white whole-wheat flour can be substituted for the whole-wheat pastry flour.
Feel free to bake in a mini donut plan instead, figuring about 11 minutes of bake time.
Adapted from a recipe by Tracy Benjamin of Shutterbean