Apple Cider Doughnuts

INGREDIENTS

• 1/4 cup confectioners' sugar
• 2 1/2 cups apple cider
• 1 large egg plus 1 egg yolk
• 1/4 teaspoon baking soda
• 1/4 cup buttermilk
• 1 2/3 cups granulated sugar
• 3 tablespoons vegetable shortening
• 1 teaspoon vanilla extract
• 1/4 teaspoon freshly grated nutmeg
• 3 1/2 cups all-pourpose flour, plus more for dusting
• 4 teaspoons baking powder
• 1 teaspoon salt
• 3 1/2 teaspoons ground cinnamon
• Vegetable oil, for frying
• 2 red apples, such as Cortland or McIntosh