Cider Mini-Doughnuts

Boil cider in a small heavy saucepan until reduced to about 1/3 cup, 12 to 15 minutes, then cool completely. Whisk together flour, baking powder, baking soda, salt, and 1 teaspoon cinnamon. Whisk together reduced cider, buttermilk, butter, eggs, and 1 cup sugar in a small bowl. Stir into flour mixture until a dough forms (it will be sticky). Turn out dough onto a well-floured surface and pat out with lightly floured hands into a 13-inch round (about 1/3 inch thick). Cut out rounds with floured cutter and transfer to a floured baking sheet. Repeat with scraps (reroll only once). Heat oil to 370 °F in a 5-quart heavy pot over medium heat. Slide 10 doughnuts into oil. When each doughnut floats to surface, turn over and fry, turning once more, until golden brown, 1 1/2 to 2 minutes. Drain on paper towels. Make more doughnuts, returning oil to 370 °F between batches. Cool slightly; dredge in cinnamon sugar made from remaining cup sugar and remaining teaspoon cinnamon.