Core the apples. Peel if desired (but not required). Coarsely chop and place in a saucepan. Add the 1 1/2 cups of apple cider and turn the heat on to medium heat. Cover the pan and cook until the apples start to soften, about 8-10 minutes.

Uncover the pan and continue to cook until the cider is reduced, about 15 minutes. Puree with an immersion blender or remove to a blender and puree until smooth. Return to the pan and the heat and continue to cook down until you have 1 cup of sauce. Cool slightly.

In a bowl, whisk together the flour, baking powder, cinnamon, salt, baking soda and nutmeg.

In another bowl, combine the sugar and the shortening and mix with a mixer on medium speed until the mixture looks sandy. Beat the egg and egg yolk into the sugar mixture.  Add the applesauce and mix, scraping the bowl as needed.

Add half of the flour mixture and mix until combined, then add the buttermilk and vanilla, mixing to combine. Add the remaining flour and mix just until combined. Do not overmix.

Place a piece of plastic wrap on a baking sheet. Pour the dough out onto the center of the plastic wrap. With floured or wet hands, pat the dough into a 7x11-inch rectangle. The dough will be sticky. Use the plastic wrap to wrap the dough, then transfer the dough on the baking sheet to the refrigerator to chill overnight.

When you are ready to make the donuts, heat 2 inches of oil in a deep pot or heavy, tall sided skillet. Heat the oil to 350ºF.

Line a baking sheet with paper towels and set aside.

In a small pot, start the glaze. Place the 1 cup of apple cider in the pot and cook over medium heat until it is reduced to about 1/4 cup. Whisk in the powdered sugar until smooth. Keep warm.

In a shallow bowl, mix together the 1/2 cup sugar and the 1 teaspoon of cinnamon.

Flour your counter or work surface, then turn the dough out onto the surface. The dough will still be sticky, so use extra flour as needed to keep it from sticking. Roll the dough until it is a 9x12-inch rectangle.

Use a 3-inch round cutter to cut 12 donuts. Use a 1-inch round cutter for the centers.

When the oil is hot, add 2-3 donuts to the hot oil at a time, frying until they are golden brown, then flipping to cook the second side. You will cook them about 2-3 minutes per side. Transfer the donuts to the paper towel lined baking sheet and continue with the remaining donuts, including the donut holes.

When the donuts are cool enough to handle, dip the tops into the glaze and then into the cinnamon sugar.

Serve the donuts warm. Store any leftover donuts lightly covered.