Combine flour, yeast, baking powder, sugar, apple pie spice, and salt in the bowl of an electric mixer.
Add apple cider, egg, and boiled cider.
Mix on medium low until combined.
Add butter and continue mixing.
Switch to dough hook, if using.
Knead dough until it is smooth and cleans the bowl, about 8-10 minutes.
Place in a covered container that has been sprayed with cooking spray.
Let rise at room temperature for 1 hour, until doubled in bulk.
On a lightly floured surface, roll dough to 1/2" thickness.
Cover and let rest for 10 minutes.
Using a 1" cutter, cut out donuts.
Use a pastry brush to brush excess flour off doughnuts and place them on a parchment lined baking sheet that has been sprayed with cooking spray.
Allow to proof for an additional 15 minutes.
While dough is proofing, heat 2 inches of vegetable oil to between 350 degrees F and 375 degrees F.
Drop donuts in one at a time, working in small batches.
Allow to cook until golden brown, 1-2 minutes.
Flip donuts and cook an additional 1-2 minutes until evenly cooked.
Remove from oil and drain on paper towels.
Roll in sugar mixture.