Preheat oven to 350°F and line a muffin tin with paper liners.
In a large bowl, whisk applesauce, brown sugar, oil, eggs, and vanilla.
In another bowl, combine flour, oats, baking soda, baking powder, salt, cinnamon, and nutmeg.
Fold the dry ingredients into the wet mixture until just combined.
Spoon batter into muffin cups about 3/4 full.
Bake 18–20 minutes, or until a toothpick comes out clean.
Cool slightly before serving.