Place the rack in the center of the oven, and then preheat to 375F. Line a 12-cup muffin pan with paper liners.
In a large bowl, whisk the flour with baking soda, baking powder, cinnamon, and salt.
Then in a medium bowl, whisk eggs in oil, buttermilk, sugars, applesauce, and vanilla.
Stir egg mixture into flour mixture.
Add apple, carrots, and rolled oats to the mixture while stirring. Do not over mix.
Scoop into lined muffin cups.
For about 18 to 20 minutes, bake until a skewer inserted into the center of a muffin comes out clean.