In a medium saucepan over medium-high heat, bring apple cider to a boil. Continue to boil until reduced to 3/4 cup, about 25 minutes. Pour into a large heatproof bowl and let cool to room temperature.
In a large bowl, whisk together flour, brown sugar, baking powder, salt, baking soda and cinnamon. Whisk butter and egg into cooled apple cider. Add dry ingredients to wet and stir until combined.
Cover bowl with plastic and refrigerate until dough is cold and firm, about 2 hours.
On a floured surface, roll chilled dough into a 1/2" thick rectangle, lightly flouring as necessary. Using a donut cutter or 3" and 1" biscuit cutters, punch out donuts. Bring scraps together and punch out as well. Place donuts and holes onto baking sheets, cover with dish towel.
Place cinnamon sugar in shallow bowl. Heat 1/2" oil in a large stockpot or Dutch oven over medium-high heat until a deep-fry thermometer registers 350℉.
Working in batches, add donuts to Dutch oven and cook until evenly golden and crispy, 1 to 2 minutes per side, adjusting heat as necessary to maintain frying temperature between 350℉ and 375℉.
Transfer to a paper towel-lined plate for 30 seconds, then roll warm donuts in cinnamon sugar mixture to coat.