Preheat oven to 350°F (175°C). Line a 12-cup muffin tin with liners or grease well. Make 10 larger muffins, or 12 snack-sized muffins.
In a medium bowl, whisk flour, baking powder, baking soda, salt, cinnamon, and nutmeg.
In a separate larger bowl, whisk eggs, applesauce, sugars, oil, and vanilla until smooth.
Add wet ingredients to dry and fold gently until just combined. Fold in the diced apples making sure not to overmix.
Divide batter evenly among muffin cups (about ¾ - full). Bake 18–22 minutes, until a toothpick inserted comes out clean. Cool in the pan for 5 minutes, then transfer to a wire rack.