Core and coarsely chop the apples (do not peel). Combine the apples with 1 1/2 cups cider in a medium saucepan over medium heat; cover and cook until softened, about 8 minutes. Uncover and continue cooking until the apples are tender and the cider is almost completely reduced, about 5 minutes.
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.)