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To make the doughnuts: Preheat the oven to 400°F. Grease a standard doughnut pan.
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Beat together the butter, oil, sugar, salt, and spices. Beat in the boiled cider and vanilla, then the egg. Scrape the bottom and sides of the bowl.
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Whisk together the baking powder, baking soda, and flour. Stir the flour mixture into the wet ingredients alternately with the milk, beginning and ending with flour.
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Spoon the batter into the pan, smoothing the tops.
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Bake the doughnuts for 10 to 12 minutes, until a cake tester or toothpick inserted into one comes out clean.
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Remove the doughnuts from the oven, wait 5 minutes, then turn out onto a rack to cool.
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To make the glaze: Place all the glaze ingredients into a saucepan.
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Cook over medium heat, stirring frequently, until the sugar is dissolved.
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Bring the mixture to a boil, cover the pan, and boil for 3 minutes
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Remove the cover and boil for a few more minutes, until the syrup reaches soft-ball stage, 240°F on a digital thermometer. Remove from the heat and cool slightly.
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Carefully dip the doughnuts in the warm syrup; reheat the syrup if it's thickened too much. If you dunk them completely, place them on a piece of greased parchment or waxed paper, to set. If you dip just the tops, place them on a rack. There's enough syrup to dip each doughnut twice; the second dip yields a glossy finish. Note: any extra syrup will hold for up to a week, covered, in the refrigerator.
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Top the doughnuts with chopped nuts while the glaze is still sticky, if desired.
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Store any leftover doughnuts, loosely wrapped, at room temperature for a couple of days; freeze for longer storage.