Step 1
Mix 1 cup sugar and 1 tablespoon cinnamon together in a resealable freezer bag.
Step 2
Bring apple cider to a boil in a saucepan; cook stirring occasionally until cider has reduced to 1/2 cup about 20 minutes. Remove from heat and cool.
Step 3
Whisk 1 cup flour 1 cup sugar baking powder 2 teaspoons cinnamon and salt together in a large bowl. Mix butter eggs egg yolk and cooled cider together in a separate bowl until smooth; stir into flour mixture. Stir remaining 2 3/4 cup flour into mixture until dough is smooth. Refrigerate dough for 10 minutes.
Step 4
Heat oil in a large pot or deep-fryer to 375 degrees F (190 degrees C).
Step 5
Turn dough onto a well-floured work surface using floured hands; pat dough into 1/2-inch thickness. Cut dough into doughnuts using a doughnut cutter or 2 round cookie cutters (1 large and 1 small). Gather scraps and re-pat dough to cut out as many doughnuts as possible.
Step 6
Carefully slide doughnuts working in batches into hot oil; fry until doughnuts rise to the surface and begin to brown 2 to 3 minutes. Flip doughnuts and continue frying until opposite side is browned about 1 1/2 minutes.
Step 7
Remove doughnuts with a slotted spoon and place on a paper towel-lined plate to cool until easily handled about 1 minute. Toss cooled doughnuts in the reserved bag of cinnamon-sugar. Bring oil back to 375 degrees F (190 degrees C) before frying next batch.