Preheat the oven to 350-degrees. Line your muffin tins with papers (or grease) and set aside.
Make the topping first - in a small bowl, mix together the oats, brown sugar and cinnamon. Set aside.
Beat together the sugar and butter until light and fluffy. Add the eggs and beat them in well. Add the dry ingredients and mix well. Finally, add in the applesauce and vanilla. Scrape down the bowl and spoon into the muffin cups (I like to use a muffin or ice scream scoop).
Sprinkle the prepared topping over each muffin, pressing gently so that the topping doesn't all fall off.
Bake for about 18-22 minutes, depending on your oven - a tester should come out clean, and the top should be golden brown. Remove the muffins from the oven and let set 5 minutes before removing to a cooling rack. Serve warm or chill in the refrigerator and serve cold - either way, these are delicious!