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, whisk together the all-purpose flour, rolled oats, granulated sugar, brown sugar, baking powder, baking soda, salt, ground cinnamon, and ground nutmeg.
In a separate medium bowl, beat the egg. Add the unsweetened applesauce, whole milk, vegetable oil, and vanilla extract. Mix until combined.
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 into the batter.
Divide the batter evenly among the 12 muffin cups, filling each about 3/4 full.
Bake in the preheated oven for 18 to 20 minutes, or until a toothpick inserted into the center of a muffin comes out clean.
Remove the muffins from the oven and allow them to cool in the pan for 5 minutes.
Transfer the muffins to a wire rack to cool completely before serving.