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Preheat the oven to 350°F. Lightly grease two standard doughnut pans. If you don't have doughnut pans, you can bake these in a standard muffin tin; they just won't be doughnuts.
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Beat together the oil, eggs, sugar, applesauce, boiled cider, vanilla, cinnamon, salt, and baking powder until smooth.
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Add the flour, stirring just until smooth.
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Fill the wells of the doughnut pans nearly to the rim; use about 1/4 cup of batter in each well. If you're making muffins, fill each well about 3/4 full; the recipe makes about 15, so you'll need to bake in two batches (unless you have two muffin pans).
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Bake the doughnuts for 15 to 18 minutes, or until a cake tester inserted into the center of one comes out clean. If you're making muffins, they'll need to bake for 23 to 25 minutes.
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Remove the doughnuts from the oven, and loosen their edges. After about 5 minutes, transfer them to a rack.
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While the doughnuts are still warm (but no longer fragile), gently shake them, 1 or 2 at a time, in a bag with the cinnamon sugar. If you've made muffins, sprinkle their tops with sugar. Allow the doughnuts or muffins to cool completely before glazing.
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To make the glaze: Mix together all of the glaze ingredients, stirring until smooth.
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Spread the doughnuts with glaze (or dip tops in glaze); return to the rack until the glaze is set.
