Boil apple cider over medium-high until reduced to 1/3 cup, about 30 minutes. Whisk in apple butter, buttermilk, and vanilla. Set aside.

In a bowl, whisk together flour, baking powder, salt, baking soda, cinnamon, nutmeg, and allspice until well combined. Set aside.

In a stand mixer, cream together butter, shortening, and sugar until lightened and fluffy, about 4 minutes. Add eggs one at a time, waiting until the first is incorporated before adding the next.

Add half of the dry ingredients to the butter mixture and mix on low until combined. Add half of the cider mixture and mix to combine. Repeat process with remaining flour mixture and cider mixture.

Line a baking sheet with parchment paper. Flour generously. Dump dough onto prepared baking sheet, flour the surface of dough generously and pat into 3/4-inch thickness. Cover with plastic wrap and freeze for 15 minutes. Prepare another baking sheet by lining it with parchment paper. Remove baking sheet from freezer and use a floured 3-inch doughnut cutter to cut out doughnuts. Transfer doughnuts to unfloured baking sheet, re-rolling scraps as needed. Cover doughnuts in plastic wrap and freeze for an additional 15 minutes.

Mix together cinnamon and sugar in a large bowl. Set aside.

Heat canola oil in a fryer or Dutch oven to 365°F. Fry 3 doughnuts at a time, allowing to cook for 3 minutes on one side, and 1 to 2 minutes on the other side, until the doughnut is evenly browned. Transfer doughnuts to paper towel-lined plate to dry. Roll doughnuts in cinnamon sugar while still warm. Fry remaining doughnuts in batches of 3, rolling each doughnut in cinnamon sugar after it has drained on the paper towels.