In a large bowl, sift together flour, sugar, baking powder, apple pie spice, and salt.
Add in apple cider, egg, butter, Greek yogurt, and vanilla. Stir with a wooden spoon until ingredients are fully incorporated. (It will be sticky.) Cover with a damp towel and refrigerate for one hour.
When ready, roll out the dough to 1/4-inch to 1/2-inch thick. (The thinner the dough, the more donuts you can make.) Try to make them as consistent in size as possible.
Once rolled out, use a donut cutter to cut out the shapes. If you do not have a donut cutter, use a round tool like a kitchen glass or a jar lid that is about 3-inches across. Use a bottle cap or small shot glass for the smaller hole.
Set the cut-out donuts on a baking sheet to let rest as you heat the oil.