Combine the flour, sugar, baking powder, and cinnamon in a bowl. Stir to combine. Add the milk, egg, oil, and beer. Mix with a fork to combine, then use your hands to mix well and bring it together into a sticky dough.

Heat a deep fryer or three-quart pot with oil, and, using oiled hands, drop small balls of dough into the hot oil. The donut holes will double in size. Fry for about four to five minutes until golden, moving them around into the pot for even frying. Let drain on a cooling rack over a baking pan, or over paper towels.

Separately, combine cinnamon and sugar on a shallow plate. While donuts are still warm, roll them in the cinnamon sugar.