In a large bowl, combine 1-1/2 cups of the flour, the yeast, and apple pie spice; set aside. In a medium saucepan, heat and stir sugar, cider, milk, butter, and salt just until warm (120 degrees F to 130 degrees F) and butter almost melts. Add to flour mixture along with eggs. Beat with an electric mixer on low to medium speed for 30 seconds, scraping sides of bowl constantly. Beat on high speed for 3 minutes. Using a wooden spoon, stir in as much of the remaining flour as you can.
Turn dough out onto a lightly floured surface. Knead in enough of the remaining flour to make a moderately soft dough that is smooth and elastic (3 to 5 minutes total). Shape dough into a ball. Place in a lightly greased bowl, turning once to grease surface; cover. Let rise in a warm place until double in size (about 1-1/2 hours).
Punch dough down. Turn out onto a lightly floured surface. Divide in half. Cover; let rest for 10 minutes. Line a large baking sheet with waxed paper. Lightly flour waxed paper; set aside.
Roll each dough half to 1/2-inch thickness. Cut with floured 2-1/2-inch doughnut cutter, dipping cutter into flour between cuts. Reroll as necessary. Place on prepared baking sheet. Cover and let rise in a warm place until very light (45 to 60 minutes). Fry doughnuts, 2 or 3 at a time, in deep, hot fat (365 degrees F) about 1 minute on each side or until golden, turning once with a slotted spoon. Drain on paper towels. Repeat with remaining doughnuts and doughnut holes.
Dip tops of cooled doughnuts in the Spice Glaze and let stand until set. Makes 16 to 18 doughnuts plus holes.
In a small bowl, combine powdered sugar and apple pie spice. Stir in enough milk to make a thin icing consistency.