Preheat the oven to 375°F (190°C). Line a 12-cup muffin tin with paper liners or grease the cups.
In a large bowl, combine the rolled oats, all-purpose flour, brown sugar, baking powder, baking soda, ground cinnamon, ground nutmeg, and salt. Stir until well mixed.
In a separate medium bowl, whisk together the applesauce, milk, vegetable oil, egg, and vanilla extract until smooth.
Pour the wet ingredients into the dry ingredients and stir gently until just combined. Do not overmix.
Fold in the finely chopped apple pieces evenly throughout the batter.
Divide the batter evenly among the 12 muffin cups, filling each about 3/4 full.
Bake in the preheated oven for 22 to 25 minutes, or until a toothpick inserted into the center of a muffin comes out clean.
Remove the muffins from the oven and let them cool in the tin for 5 minutes.
Transfer the muffins to a wire rack to cool completely before serving.