Bring 3 ½ cups apple cider to a low boil and reduce to 1 cup, stirring occasionally. Divide.
In small bowl, add 1/2 cup of reduced cider to finely chopped apple. Let sit while you make the doughnut batter.
For the batter:
Mix butter into sugar with clean hands until it resembles wet sand. Add egg and whisk until very pale in color. In separate bowl, whisk flour, baking powder, soda, salt and nutmeg together. Add buttermilk, vanilla & apple mixture to butter/sugar/egg mixture. Add flour mixture just until combined. Generously flour a piece of parchment, and pat dough out onto parchment about ½-3/4” thick. Dust top with flour, cover, and refrigerate until cold—preferably overnight.
After dough is chilled, heat oil to 360 degrees F. While the oil is heating, mix sugar and cinnamon for dipping—we recommend using a pie plate. Cut out or shape doughnuts and/or holes as desired. Fry one side until golden brown, then flip and cook another minute or two. Let cool on rack a minute or two, then roll in cinnamon sugar.
Serve with caramel sauce on side for dipping or drizzling!
For caramel sauce:
Combine ¾ cup sugar and 1/3 cup reduced cider in medium saucepan over medium high heat. Boil until color changes to a darker brown and syrup-y (caramel colored). Carefully add cream (it will bubble at you—be careful!). Bring back to boil, and cook a few minutes. Cool slightly, then pour on everything.