Apple Cider Glazed Yeast Doughnuts

1. In a large bowl, combine the yeast, salt, 1/3 cup sugar, and 1 1/2 cups flour. Warm the butter and 1 cup water in the microwave until it registers 120 degrees F to 130 degrees F on a thermometer. Add the butter mixture to the flour mixture. Using an electric mixer, beat until smooth, about 2 minutes.
2. Reduce the mixer speed to low and beat in the eggs, vanilla, and zest until blended. Gradually add the remaining 2 1/4 cups flour (1/2 cup at a time), mixing until just combined. Beat on medium speed for 5 minutes.
3. Coat a large bowl with 1 tablespoon canola oil. Transfer the dough to the bowl. Cover with plastic wrap and refrigerate overnight.
4. The next day, line 2 baking sheets with parchment paper. Coat the parchment with 2 tablespoons oil. On a lightly floured surface, punch down the dough and knead a few times. Let rest for 5 minutes. Roll the dough into a 3/4-inch-thick circle. Using the 3 1/4-inch cutter, cut out 7 rounds and transfer to the prepared baking sheets, spacing them about 2 inches apart. Using the 1 1/2-inch cutter, cut out the centers. Gather the doughnut holes and scraps and knead together. Reroll and cut the scraps until you have 12 doughnuts. Cover with oiled parchment or plastic wrap and let rise in a warm place until nearly doubled in size, 20 to 30 minutes.
5. Meanwhile, line baking sheets with paper towels. Heat the remaining 4 cups oil in a large skillet until a deep-fry thermometer registers 350 degrees F.
6. Working in batches, lower doughnuts into the hot oil. Fry, turning occasionally, until golden brown, 1 to 2 minutes total. Transfer to the prepared sheets to drain. Transfer doughnuts to a wire rack over a baking sheet and allow to cool completely.
7. Meanwhile, in a medium bowl, stir together confectioners’ sugar, cider, and apple pie spice until smooth. Dip top of doughnuts into glaze, letting excess drip off, and place on wire rack until glaze has set, 10 minutes.