Instructions:
In a large bowl using a hand-held or standing mixer fitted with a whisk attachment, beat together sugar and butter until mixture is pale and fluffy, 4-6 minutes. Add eggs, one at a time, beating a minute after each. In a medium-size bowl, whisk together flour, salt, baking powder, baking soda, cinnamon, and nutmeg; set aside.

Pour buttermilk, boiled cider, and vanilla into sugar/butter/egg mixture. Mix well, and don't worry if the mixture looks a bit curdled; it'll smooth itself out. Add flour mixture and combine gently just until fully moistened.

Line two baking sheets with waxed paper or parchment paper and dust generously with flour. Turn dough out onto one baking sheet and pat gently into 3/4-inch-thickness. Sprinkle dough with additional flour, cover with plastic wrap, and place in the freezer for 10 minutes to firm up. Remove dough from the freezer; use a lightly floured 3-inch doughnut cutter (or two concentric biscuit cutters) to cut out about 18 doughnuts with holes. (You may gather the scraps and roll again as needed, but you may need to chill the dough more to firm it up.) Place cut doughnuts on the other baking sheet as you go; then transfer to the freezer for 5 minutes to firm up again.

Line a plate with a few layers of paper towels and set it nearby. In a Dutch oven or large pot, heat 3 inches of oil to 370° (test with an instant-read thermometer). Drop 3 or 4 doughnuts into the oil, being careful not to crowd the pan. Cook until browned on one side, about 1 minute; then flip and cook until browned on the other side, about 1 minute longer.

Repeat with the remaining dough (if you find that it's getting too soft as you work your way through the batches, pop it into the freezer again for 10 minutes). When doughnuts are cool enough to handle but still warm, sprinkle all over with cinnamon sugar or confectioners' sugar. Serve immediately.

Note: Boiled apple cider gives these apple cider donuts a rich, slightly tangy flavor. You can buy boiled cider at some gourmet and Whole Foods stores; from Wood's Cider Mill in Springfield, Vermont or from the King Arthur Flour catalogue. Alternatively, you can boil your own cider by simmering 1-1/2 cups of fresh apple cider down to 1/3 cup in about 25 minutes--it just won't be as concentrated as the commercial product.