Mix 1 cup of sugar and 1 tablespoon of cinnamon together in a resealable plastic bag.

Bring the apple cider to a boil in a saucepan and cook, stirring occasionally, until reduced to 1/2 cup. This takes around 20 minutes. Remove from heat and allow to cool.

In a large bowl, whisk together 1 cup of flour, 1 cup of sugar, baking powder, 2 teaspoons of cinnamon, and salt. Separately, mix butter, eggs, egg yolk, and cooled cider until smooth. Combine with the dry ingredients and stir until smooth. Incorporate the remaining 2 3/4 cups of flour. Refrigerate the dough for 10 minutes.

Heat oil in a deep fryer or large pot to 375°F (190°C).

On a well-floured surface, pat the dough to a thickness of 1/2 inch. Use a doughnut cutter or two round cookie cutters (different sizes) to cut out doughnuts. Gather scraps and repeat the process.

Carefully place the doughnuts into the hot oil in batches. Fry for 2-3 minutes until one side rises to the surface and browns. Then flip and fry the other side for 1 1/2 minutes.