1) 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.
2) Beat together the oil, eggs, sugar, puréed fruit, vanilla, cinnamon (if you're using it), salt, and baking powder until smooth.
3) Add the flour, stirring just until smooth.
4) 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).
5) 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.
6) Remove the doughnuts from the oven, and loosen their edges. After about 5 minutes, transfer them to a rack.
7) While the doughnuts are still warm (but no longer fragile), gently shake them, 1 or 2 at a time, in a bag with the sugar. If you've made muffins, sprinkle their tops with sugar.
8) Cool completely, and wrap airtight; store at room temperature for several days. Freeze for longer storage.
9) To make dessert doughnuts: Fill the hole in each doughnut with your choice of ice cream, pudding, mousse, sliced fruit, etc. Top with sauce; add whipped cream (and nuts, and a cherry) if desired.
Yield: 12 doughnuts or 15 muffins.