1) Preheat the oven to 325°F. Lightly grease the top and bottom parts of a doughnut hole pan. If you don't have a doughnut hole pan, a mini muffin pan will work well; see "tips from our bakers," at left below.
2) In a medium-sized mixing bowl, beat together the butter, vegetable oil, and sugars until smooth.
3) Add the eggs, beating to combine.
4) Stir in the baking powder, baking soda, salt, nutmeg, and vanilla.
5) Stir the flour into the butter mixture alternately with the milk, beginning and ending with the flour and making sure everything is thoroughly combined.
6) Spoon the batter evenly into the prepared pan, filling the cups full; the batter will mound up a tiny bit. Place the top on the pan, and lock it in place with the clips.
7) Bake the doughnut holes for 15 to 17 minutes, or until they're a pale golden brown and a cake tester inserted into the middle of one of the center holes comes out clean.
8) Loosen the edges of the holes if necessary, and gently return them onto a rack to cool briefly. Grease the pan again, and bake as many more batches as necessary to use up the batter.
9) Place the cinnamon-sugar in a paper bag, and add the doughnut holes a few at a time. Shake gently to coat the holes with the cinnamon-sugar. If the holes have cooled too much and aren't holding the sugar, spritz them very lightly with a bit of water, then shake.
10) Serve doughnuts warm or at room temperature. Store for several days at room temperature, lightly covered; freeze for longer storage.
Yield: about 40 doughnut holes.