In the bowl of a food processor, add the oats. Pulse the oats about 20 times to chop them up, but don't turn the oats into flour.
Add the all purpose flour, baking powder, baking soda, salt, and cinnamon to the food processor bowl with the oats. Pulse 5 or 6 times to get all the dry ingredients combined. Set aside.
Melt the butter in a separate bowl until just melted but not hot, and set aside.
In a large bowl, whisk together the applesauce and sugar.
Add in the egg and the vanilla and whisk until well incorporated.
Drizzle the melted butter into the applesauce and egg mixture. Whisk while you drizzle to evenly distribute the butter into the wet ingredients.
Pour the flour mixture into the wet ingredients. Using a rubber spatula or wooden spoon, stir the batter until almost fully mixed. Leave the batter lumpy with streaks of flour showing.
Add the grated apple and carrots to the lumpy batter. Fold the carrots and apples into the batter until evenly distributed and no flour streaks remain. Do not overmix.