Mix 1 cup sugar and 1 tablespoon cinnamon together in a resealable freezer bag.
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.
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.
Heat oil in a large pot or deep-fryer to 375 degrees F (190 degrees C).
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.
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.
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.
