• 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, baking soda, 2 teaspoons cinnamon, and salt together in a large bowl. Add melted butter, eggs, egg yolk, and cooled cider together to flour mixture. Stir remaining 2 1/2 cups 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 place doughnuts, working in batches, into hot oil; fry until doughnuts rise to the surface and begin to brown, about 2 minutes. Flip doughnuts and continue frying until opposite side is browned for another minute or so.
• Remove doughnuts with a slotted spoon and place on a paper towel-lined plate to cool until easily handled, about 1 minute. Bring oil back to 375 degrees F before frying next batch.

Cinnamon Sugar Version

• Mix 1 cup sugar and 1 tablespoon cinnamon together in a resealable freezer bag.
• Toss warm doughnuts in the bag of cinnamon-sugar.

Buttercream Caramel Version

• Heat butter and honey in microwave until melted. Stir. Add milk and vanilla. Stir in powdered sugar.
• Dip cooled donuts in buttercream glaze.
• Drizzle in caramel dip.