Write instructions - See below
To make the dough, combine the flour, baking powder, baking soda, salt and spices in a medium bowl. Whisk to blend, and set aside. In the bowl of an electric mixer, combine the sugar and butter. Beat until well blended. Scrapping down the sides of the bowl to make sure it is all combined. Stir in the egg, egg yolks, and then the vanilla until incorporated.
In a large liquid measuring cup combine the buttermilk and pumpkin. Whisk until smooth. With the mixer on low speed, add the dry ingredients in three additions alternating with the pumpkin mixture, beginning and ending with the dry ingredients. Once the dough is mixed, cover and chill for at least 3 hours or until firm.
On a well-floured work surface, gently knead the dough 2-3 times, roll or pat out the dough to a ½=inch thick round. Sprinkle the surface of the dough with flour. Using a 2½ to 3-inch round biscuit cutter, cut out the rounds of dough. Use a smaller cutter (or a wide pastry tip) to cut a hole out of the center. Reroll and cut the dough scraps as needed.
Add oil to a large Dutch oven or sauce pan to a depth of about 2-3 inches. Attach a thermometer to the side of the pan and heat the oil to 365-370°F. Carefully Add the rings of dough to the hot oil so that they are in a single layer and not touching. Fry, turning once, until both sides are golden brown and doughnuts are cooked through, about 2-3 minutes. Use a skimmer/strainer to remove from the oil and transfer to a paper towel-lined rack or plate. Bring the oil temperature back up to the target range before repeating with the next batch of doughnuts. Use the same process for the doughnut holes, frying for a shorter time.
To make the cinnamon-sugar, combine the sugar and cinnamon in a shallow dish and whisk to blend. When the doughnuts are just cool enough to handle, dip half of the batch in the cinnamon-sugar to coat completely, shaking off the excess.
To make the spiced glaze, combine the powdered sugar and spices in a small bowl. Add the milk and whisk to combine, until a thick glaze is formed. If necessary, add a bit more milk to thin the glaze out. Dip the remaining doughnuts in the glaze. Set on a cooling rack and allow glaze to set before serving.