To make the doughnut dough: Sift the flour, baking powder, salt, nutmeg, and pumpkin pie spice together into a medium bowl, and set aside.

In a stand mixer fitted with the paddle attachment, mix the sugar and shortening/vegetable lard for 1 minute on low speed, until sandy. Add the egg yolks, then mix for 1 more minute on medium speed, scraping the sides of the bowl with a rubber spatula if necessary, until the mixture is light colored and thick.

Add the dry ingredients to the wet ingredients in three separate additions, alternating with the sour cream and pumpkin, mixing until just combined on low speed and scraping the sides of the bowl each time. The dough will be sticky, like wet cookie/biscuit dough.

Transfer the dough to a clean bowl and refrigerate, covered with plastic wrap/cling film, for 45 minutes (or up to 24 hours).

Meanwhile, make the pumpkin glaze: Place the confectioners'/icing sugar, corn/golden syrup, salt, pumpkin pie spice, pumpkin, and vanilla in the work bowl of a stand mixer fitted with the paddle attachment. With the machine on medium speed, add the water in a slow, steady stream, and blend until all of the sugar has been incorporated, scraping the bowl a few times if necessary. Set aside.

Using a candy thermometer to measure the temperature, heat oil (at least 2 in deep) in a deep fryer, large pot, or high-sided frying pan to 325°F. Roll out the chilled dough on a generously floured counter or cutting board to 1/2 in thick, or about 8 inches in diameter, flouring the top of the dough and the rolling pin as necessary to prevent sticking. Cut into as many doughnuts and holes as possible, dipping the cutter into flour before each cut. Fold and gently reroll the dough to make extra holes (working with floured hands makes the dough less sticky), and cut again.

Shake any excess flour off the doughnuts before carefully adding them to the hot oil a few at a time, taking care not to crowd them. Once the doughnuts float, fry for 15 seconds, then gently flip them. Fry for 75 to 90 seconds, until golden brown and cracked, then flip and fry the first side again for 60 to 75 seconds, until golden. Transfer to a rack set over paper towels/absorbent paper.

While the doughnuts are still quite hot, dip the side with the deepest cracks on each into the warm Pumpkin Glaze. Let dry on cooling racks, glazed side up, for about 15 minutes.