Core and coarsely chop apples, but do not peel. Put in saucepan and add 1 1/2 cups cider, cover and cook over medium heat for 8 minutes until softened. Uncover and continue cooking for 5 minutes until apples are tender and the cider is almost completely reduced. Puree until smooth. Measure the sauce, you need 1 cup; boil it longer to reduce more if you need to. Cool slightly.
Whisk flour, baking powder, baking soda, 1 1/2 t. cinnamon, salt, and nutmeg together in a bowl.
In another bowl, beat 2/3 cup sugar and shortening with mixer on medium speed until sandy. Beat in eggs, then gradually add the pureed apple mix. Beat in half the flour mixture, then buttermilk and vanilla, then rest of flour. Mix to make sticky dough, but don't overmix.
Scrape dough onto a lightly floured sheet of parchment paper and pat into 7x11 inches, about 1/2 inch thick. Cover with plastic wrap and refrigerate at least 2 hours or overnight.
Meanwhile, make the glaze by simmering 1 cup cider over medium heat until reduced to 1/4 cup. Whisk in powdered sugar until smooth and glossy, set aside. Mix 1 cup sugar and 2 t. cinnamon in shallow bowl, set aside for topping.
Heat 2 inches of oil in large heavy-bottomed pot over medium-high heat until reaches 350 degrees. Line a baking sheet with paper towels. Cut the chilled dough with a floured 2 1/2-3 inch biscuit cutter, then cut out the middle with a 1 inch cutter ( or use a doughnut cutter).
Fry 2-3 doughnuts at a time until golden brown, 1-2 minutes per side, transferring to paper towels when done. Fry the dough scraps as doughnut holes.
Dip one side of each doughnut in the cider glaze, letting excess drip off, then dip in the cinnamon-sugar. Serve warm.