Preheat oven to 400 degrees Fahrenheit and spray with oil or butter a 24 count donut hole pan or a mini muffin pan.
In a large bowl, combine flour, sugar, baking powder, baking soda, cinnamon, nutmeg, and salt. Add milk, egg, applesauce, and vanilla and stir till just combined, making sure not to overmix. A few lumps can remain. Spoon the batter evenly into 24 mini muffin cups, about 2 tablespoons each.
Bake at 400 degrees F for 7-9 minutes, till just puffed up and very lightly browned on the edges.
Let the donut holes cool for 5 minutes then gently remove them from the pan and place them onto a cooling rack with a cookie sheet underneath. This will catch all of the drippings from the toppings.
Set aside the melted butter, and whisk sugar and cinnamon in another bowl. In a third bowl, whisk powdered sugar, milk, and vanilla till smooth. For the cinnamon sugar donut holes, dip them in the melted butter and then roll them into the cinnamon sugar to coat evenly. Set on the baking rack. Dip the donut holes into the glaze or drizzle it on top.