Pour the mixture into a food processor, and puree until smooth (or use an immersion blender). Pour the applesauce into a measuring cup. If you have more than 1 cup, return the applesauce to the saucepan, and boil until reduced to 1 cup. (I had to do this step). Let cool slightly.

While the apples are cooking, whisk the flour, baking powder, baking soda, 1 1/2 teaspoons cinnamon, salt, and nutmeg in a medium bowl.

In bowl of electric mixer fitted with the paddle attachment, combine 2/3 cup granulated sugar and the shortening, and beat on medium speed until sandy.

Beat in the egg and yolk, and then gradually mix in the applesauce, scraping the bowl. Mix in half of the flour mixture, then the buttermilk and vanilla, and then the remaining flour mixture. Mix to make a sticky dough; do not overmix.

Scrape the dough onto a lightly floured sheet of parchment paper laid on a sheet pan turned upside down. Roll dough in the flour to lightly coat (I added this step, as the dough stuck to the parchment so perhaps this will help). Pat dough into a 7-by-11-inch rectangle, about 1/2 inch thick. Cover with plastic wrap and refrigerate for at least 2 hours or overnight. (Overnight is recommended because the colder the dough is, the easier it is to work with. It's a very sticky dough.)

When you're ready to make the doughnuts, heat 2 inches of vegetable oil in a large heavy-bottomed pot over medium-high heat until a deep-fry thermometer registers 350 degrees.

Line a baking sheet with paper towels, and set it near the stove.

Cut the chilled dough into rounds, using a floured-dipped 3-inch biscuit cutter. Cut out the centers of the doughnuts, using a 1- or 1.5-inch biscuit cutter. Or simply use the small biscuit cutter to cut all doughnut holes. (The dough expands when it hits the oil, so I ended up halving the doughnut holes made with a 1.5-inch biscuit cutter). At this point, I found it useful to lightly dust/roll the doughnuts in flour so they weren't so sticky. This really helped and didn't affect the frying process or the resulting taste of the doughnuts.

Using a skimmer, tongs, or your hands (be very careful!!!), slip 2 or 3 doughnuts or 4 or 5 doughnut holes at a time into the hot oil and fry until golden brown, 1 to 2 minutes per side for regular doughnuts, approximately 3 1/2 - 5 minutes for doughnut holes (this will depend on the temperature of the oil. I had a hard time regulating it with my electric range...perhaps gas is better for this?) I suggest starting out with one doughnut hole and seeing how long it takes to get the center totally cooked. This will give you a good idea of the outside color when the inside is cooked. Our first doughnut hole was gooey on the inside, but every other doughnut after that was perfectly cooked.

Transfer to the paper towels to drain. If you prefer to eat the doughnuts hot and fresh (like we did!), only allow the doughnuts to drain for a minute or so, then drop a doughnut into the cider glaze, roll around using tongs, and then transfer to cinnamon-sugar mixture and roll around to coat entirely. The original recipe suggests to just dip one side of each doughnut, but we like them totally coated in cinnamon-sugar!

Repeat with remaining dough. Check the oil temperature periodically, and adjust the heat as needed.