Preheat the oven to 375 degrees F and grease a 24-cup mini muffin pan with nonstick spray. (I do not recommend using paper liners with this recipe as they are likely to stick.)
Add the applesauce, milk, butter, egg, and vanilla to a medium bowl. Whisk to combine well.
Add the flour, sugar, cinnamon, baking powder, baking soda, and salt to the bowl. Whisk to combine into a thick batter.
Divide the batter among the prepared muffin tin, using about 1½ tablespoons batter in each cup.
Bake for 10-12 minutes, or until puffed up, golden brown, and a cake tester or toothpick inserted into the center of a muffin comes out clean.
Remove from oven and let cool for a minute or two. Run a knife around the edges to loosen, as needed, then transfer muffins to a wire rack to cool completely.
Optional: Stir the sugar and cinnamon together in a small bowl. Dip the top of each muffin first in the melted butter, then in the cinnamon-sugar mixture. This makes them taste like a cake donut hole!