Preheat the oven to 375°F (190°C). Line a 12-cup muffin tin with paper liners or grease it lightly.
In a large bowl, whisk together the gluten free all-purpose flour, baking powder, baking soda, salt, ground cinnamon, and ground nutmeg until well combined.
In a separate medium bowl, whisk together the granulated sugar, brown sugar, egg, unsweetened applesauce, apple cider, vegetable oil, and vanilla extract until smooth.
Pour the wet ingredients into the dry ingredients and gently fold together using a spatula 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 18-20 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 pan for 5 minutes. Then transfer the muffins to a wire rack to cool completely.