Sprinkle yeast over water and let stand 5 minutes, or until foamy.
In large bowl of electric mixer, 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 makd dough smooth and elastic.
Place the dough in a greased bowl, and cover. Set in a warm place to rise until double. Dough is ready if you touch it, and the indention remains.
Turn the dough out onto a floured surface, and gently roll out to 1/2-inch thickness. Cut with a floured doughnut cutter. Let doughnuts rise again until double in size. Cover loosely with a cloth.
While doughnuts rise, make glaze. In a medium saucepan, cook apple cider until the mixture is reduced to about 2/3 cup. In a small bowl, combine apple cider reduction, confectioners sugar and apple pie spice. Whisk until smooth.
Heat oil 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 towels.
Dip doughnuts into the glaze while still hot, and set onto wire racks to drain off excess. Keep a cookie sheet or tray under racks for easier clean up.
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