Whisk egg and yolk together in a large bowl until frothy. Add pumpkin, sugar, buttermilk, brown sugar, and butter; whisk until combine. Stir in baking powder, pie spice, salt and soda.
Fold flour in gradually until a sticky dough forms; cover with plastic wrap and freeze 15 minutes. Meanwhile, heat 2" oil to 370 degrees in an electric skillet, deep fryer, or straight-sided pan over medium heat.
Pat out dough on a well-floured surface to 1/2 inch thick, then cut with 3 inch and 1 inch biscuit cutters dipped in flour. Transfer doughnuts and holes to a floured baking sheet using a spatula that's been dusted in flour.
Fry doughnuts and holes in batches in the oil until browned, about 3 minutes, turning once (holes will cook faster). Drain on a paper towel-lined baking sheet. Finish with sugar or apple cider glaze.
Apple Cider Glaze (Makes 3/4 cup)
1/4 cup apple cider, pinch of salt, 3 cups powdered sugar. Heat cider and salt in small saucepan over low until hot. Off heat, whisk in powdered sugar until smooth. Dip cooled doughnuts in glaze and let stand about 5 minutes.
Cinnamon Sugar (Makes 1 cup)
1 cup sugar, 1 teaspoon ground cinnamon. Mix sugar and cinnamon in a small paper bag. Toss donuts and holes in sugar mixture while hot.