In a medium bowl, sift together the flour, baking powder, salt, nutmeg, and pumpkin pie spice. Set aside.

In a large bowl, use an electric mixer to beat the sugar and butter on low speed until sandy, about 1 minute. Add the egg yolks, increase the speed to medium, and beat until light and thick, about 1 minute.

On low speed, gradually add the flour mixture to the bowl in three additions, alternating with the sour cream and pumpkin, mixing until combined. The dough should be sticky like biscuit dough.

Transfer the dough to a clean bowl and cover with plastic wrap. Refrigerate for 45 minutes, or up to 24 hours.

In a large bowl, use an electric mixer to beat the powdered sugar, corn syrup, salt, pumpkin pie spice, pumpkin puree, and vanilla until combined. On medium speed, add gradually add the water, scraping down the bowl, until incorporated. Set aside.

When ready to fry, pour 2 inches of oil into a large heavy pot. Attach a candy thermometer and heat oil to 325°F.

On a generously floured work surface, roll out the cold dough to 1/2-inch thick 8-inch wide round. Cut as many doughnuts and holes as possible, dipping the cutter into flour to prevent sticking. Fold and gently re-roll the scraps to make extra holes.

Shake any excess flour off the doughnut dough before adding them to the hot oil about three at a time. Wait for the doughnuts to float, then fry for 20 to 30 seconds. Gently flip over and continue frying for about another minute, or until golden. Transfer to a paper towel lined cooling rack to drain. Fry the holes for about 20 to 30 seconds per side.

While the doughnuts are still hot, give the glaze a whisk before dipping the cracked side of each doughnut into the glaze. Let dry on the cooling racks until set, about 15 minutes, before serving. Glazed doughnuts should be served the same day.