In a small bowl, sprinkle yeast over water and let stand 5 minutes or until foamy.
In large bowl of stand mixer with paddle attachment, cream sugar and shortening. Add yeast mixture, milk, salt, apple pie spice, eggs and 2 cups of the flour.
Beat in remaining flour 1/2 cup at a time, until the dough no longer sticks to the bowl. Remove from bowl and knead for about 5 minutes, adding flour if necessary to make dough smooth and elastic.
Place the dough in a greased bowl, and cover with plastic wrap. Set in a warm place to rise until double. Dough is ready if you touch it and the indentation remains.
Turn the dough out onto a floured surface, and gently roll out to 1/2-inch thickness. Cut with a floured doughnut cutter. Cover loosely with a cloth. Let doughnuts rise again until double in size, about 1 to 2 hours.
While doughnuts rise, make glaze. In a medium saucepan, cook apple cider until the mixture is reduced to about 2/3 cup. In a large bowl, combine apple cider reduction, confectioners sugar, corn syrup and apple pie spice. Whisk until smooth.
Heat oil in deep fryer to 375°F. Slide doughnuts into the preheated oil two at a time. Fry each doughnut about 1 minute per side. Remove from hot oil and drain on paper towel lined wire racks.
Dip doughnuts into the glaze while still warm, and set onto wire racks to drain excess. Keep a cookie sheet or tray under racks for easier clean up. Repeat dipping to double glaze doughnuts.