1. Line baking sheet with parchment. Line a large plate with paper towels. In a paper bag, combine 1 cup sugar and 1 Tbsp. cinnamon; shake to mix well. Set aside.
2. Bring cider to a boil over medium heat. Cook, stirring occasionally, until cider has reduced to 1/2 cup, about 20 minutes. Let cool to room temperature.
3. In a large bowl, whisk baking powder, salt, 1 cup flour and remaining sugar and cinnamon until well combined. In a small bowl, whisk butter, eggs, yolk and cooled cider until well combined. Add cider mixture to flour mixture; whisk until smooth. Add remaining 2 1/2 cups flour and mix until just combined. Dough will be slightly sticky. Refrigerate 10 minutes.
4. Pour oil into a large pot and attach a candy thermometer to side (thermometer should not touch bottom of pot). Heat oil over medium-high heat until it reaches 375ºF, watching and adjusting as needed to keep oil from overheating.
5. Turn dough out onto a well-floured countertop and coat fingertips with flour. Pat dough out to 1/2-inch thickness. Using a doughnut cutter or 2 round cookie cutters (one large and one small), cut out doughnuts and holes. Transfer both to lined baking sheet. Gather scraps and re-pat dough, cutting as many doughnuts as possible.
6. Working in batches, carefully slide doughnuts into hot oil. After about 1 minute, doughnuts will rise to surface and begin to brown. At that point, turn over and fry 1 1/2 minutes longer. Transfer to lined plate to drain. Bring oil back to 375ºF before frying another batch. When doughnuts have cooled for 1 minute, add to bag with cinnamon sugar and toss to coat. Remove to a clean plate and serve.