Preheat the oven to 375 degrees F and grease a 12-cup 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, oats, 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 ¼ cup batter in each cup.
Bake for 14-16 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.
Notes
Store cooled muffins in an airtight container at room temperature for up to 3 days or about a week in the fridge.
Dairy-free: Use vegetable, avocado, or coconut oil in place of the butter and nondairy milk.
Gluten-free: Use an all-purpose gluten-free flour.
Egg-free: Omit the egg. Add 2 tablespoons additional milk. Proceed with the recipe. (The muffins may be a little more tender than with the egg.)
You can lower the sugar in the muffins to 1/4 cup, though I have found that the 1/3 cup listed in the recipe is my preference for both flavor and texture.
This recipe has the best texture when made with granulated sugar, rather than maple syrup or honey.