Preheat the oven to 375°F (190°C). Line a 12-cup muffin tin with paper liners or grease the cups lightly.
In a large bowl, whisk together the gluten-free flour, baking powder, baking soda, salt, ground cinnamon, and ground nutmeg until well combined.
In a separate medium bowl, beat the eggs with granulated sugar and brown sugar until the mixture is smooth and slightly fluffy.
Add the unsweetened applesauce, vegetable oil, and vanilla extract to the egg mixture and whisk until fully incorporated.
Gradually add the dry ingredients to the wet ingredients, alternating with the milk, beginning and ending with the dry ingredients. Stir gently until just combined; do not overmix.
Fold in the chopped apples 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 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 pan for 5 minutes, then transfer to a wire rack to cool completely.