Prepare a doughnut hole pan or mini muffin pan with non-stick cooking spray.
Cream together the oil, butter, sugar, brown sugar and egg. Add the vanilla and the milk; stir. Sift together the dry ingredients and add them gradually to the butter mixture. Combine thoroughly.
Fill the prepared tins ⅔ full with batter. Bake for 10-12 minutes.
Cinnamon-Sugar Coating
While the doughnuts are baking, melt the butter for the topping and set aside. Add the sugar and cinnamon to a lunch-size paper bag.
While the doughnuts are still warm, dip first in the melted butter and then place 4 or 5 at a time in the paper bag with cinnamon-sugar. Shake the doughnuts, and place on a metal rack to cool... if you can wait that long!
Additionally, I like to make a few coated in powdered sugar.
Powdered Sugar Coating
Place ½ cup of confectioners' sugar in a paper lunch-size bag. While the doughnuts are still warm, place 4-5 in the bag, close the top, and shake.
Notes
Often when I make these doughnut holes I skip the butter coating step on the Cinnamon-Sugar coating. It's only a preference because the cinnamon sugar does stick better by first dipping them into the melted butter.