How to Make It
In a small saucepan, combine 1/4 cup plus 2 tablespoons of the fresh apple cider and the butter and bring to a boil over moderate heat. Stir in 1/4 cup plus 1 tablespoon of the flour and cook, stirring constantly, until a paste forms and pulls away from the pan, 1 to 2 minutes. Scrape the cider paste into a small bowl and let cool, then refrigerate until well chilled, about 2 hours.
In a blender, combine the cider paste with the cream, granulated sugar, eggs, vanilla bean paste and the remaining 1 cup of fresh apple cider and puree until smooth.
In a large bowl, whisk the remaining 4 1/4 cups of flour with the baking powder, salt and baking soda. Add the wet ingredients and, using a rubber spatula, stir until a sticky dough forms. Scrape the dough out onto a large baking sheet lined with plastic wrap. Cover with another sheet of plastic wrap and pat the dough 1 inch thick. Refrigerate until well chilled, at least 4 hours.
In a small saucepan, combine the granulated sugar with 1 tablespoon of water and cook over moderately high heat, swirling occasionally, until an amber caramel forms, 5 to 7 minutes. Remove from the heat and carefully add the cream and boiled apple cider syrup; the caramel will seize. Cook over moderate heat, stirring with a wooden spoon, until the caramel dissolves, about 2 minutes. Gradually add the butter and cook until thickened slightly, about 2 minutes. Stir in the salt. Scrape the mixture into a medium bowl and let cool completely. Whisk in the buttermilk and confectioners’ sugar until the glaze is smooth.
In a medium bowl, whisk the granulated sugar with the cinnamon and cardamom.
Remove the top sheet of plastic wrap and invert the dough onto a floured work surface. Peel off the plastic wrap and dust the dough with flour. Roll out the dough 1/2 inch thick. Using a 3-inch round biscuit cutter, stamp out 13 rounds. Using a 1-inch round cutter, stamp out the center from each round. Cut the scraps into 2-inch pieces. Transfer the doughnuts, holes and scraps to 2 baking sheets and refrigerate until chilled, about 30 minutes.
In a large saucepan, heat 3 inches of canola oil to 375° and top 2 baking sheets with racks. Keep the doughnuts chilled. Add one-third of the scraps at a time to the hot oil and fry, turning once, until browned, about 2 minutes per batch. Using a slotted spoon, transfer the scraps to a rack. Let the oil return to 375°. Add the doughnut holes and fry, turning occasionally, until browned all over, 1 to 2 minutes. Using a slotted spoon, transfer to a rack. Toss the doughnut scraps and holes in the cinnamon-cardamom sugar until coated and transfer to a platter.
Return the oil to 375°. In batches, fry the doughnuts, turning once, until browned, about 3 minutes per batch. Using a slotted spoon, transfer the doughnuts to a rack to cool for 10 minutes. Stir the glaze. Dip one side of each doughnut in the glaze and let stand until set before serving.
Boiled cider syrup is made by boiling apple cider until it is thick and pourable. It is available at kingarthurflour.com.