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., Beat together the oil, eggs, sugar, applesauce, boiled cider, vanilla, cinnamon, salt, and baking powder until smooth., Add the flour, stirring just until smooth., 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)., 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., Remove the doughnuts from the oven, and loosen their edges. After about 5 minutes, transfer them to a rack., 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., To make the glaze: Mix together all of the glaze ingredients, stirring until smooth., Spread the doughnuts with glaze (or dip tops in glaze); return to the rack until the glaze is set.