Place the 1/4 cup of warm cider in the bowl of a stand mixer fitted with a dough hook. Sprinkle the yeast on the surface of the cider and let stand for about 5 minutes, until foamy.
Add the flour, the remaining 1/4 cup of cider, milk, butter, egg yolks, the 2 tablespoons of sugar and the salt. Beat on the lowest speed until the flour is incorporated, then increase the speed to medium and beat for about 5 minutes to form a soft, elastic dough that creates a ball around the dough hook.
(Alternatively, you can mix the dough by hand with a wooden spoon until it forms a wet, sloppy mass. Oil your hands and give the dough a few turns using the heel of your palm against the edge of the bowl.).
Use a little of the oil to lightly grease the inside of a large bowl and place the dough in the bowl. Cover with a clean dish towel and place in a warm spot. Let the dough rise for about 1 1/2 hours, until it has doubled in bulk.
Flour a clean work surface. Line a baking sheet with parchment paper and have a wire rack ready. Heat enough oil to fill about 3 inches in a large, heavy-bottomed pot over high heat.
Punch down the dough; it will be sticky. Place the dough on the floured surface, then roll it out to a thickness of 1/2-inch, adding flour as needed to keep the dough workable. Use a 3-inch doughnut cutter or two cookie cutters, in 3-inch and 1/2-inch sizes, to create 8 to 10 doughnuts. Do not reroll the dough; because it is leavened, it needs the rising action for structure. Rerolling will flatten the lift you've achieved. Save any scraps for frying.
Place the doughnuts on the prepared baking sheet. Cover with the clean towel and let the doughnuts and any scraps of dough rise for 15 to 30 minutes, until well puffed (they do not need to double in bulk).
Combine the remaining 2 cups of sugar and the cinnamon in a medium bowl.
Once the oil reaches 325 degrees, begin frying the raised doughnuts in batches of 2 or 3 at a time. Fry for 2 to 3 minutes on the first side until golden brown, then use tongs or long metal skewers to turn the doughnuts and fry on the second side for 2 to 3 minutes until golden brown.
Use a skimmer or skewer to transfer one doughnut at a time to the bowl of cinnamon sugar; immediately turn as needed to coat evenly on all sides. Transfer to the wire rack and repeat with the remaining raised doughnuts and any leftover scraps of risen dough.