Preheat oven to 350°F. Grease a 9x5-inch loaf pan. In a large bowl, mix together flour, both sugars, baking powder, baking soda, salt, nutmeg, and cinnamon.
In a separate bowl, whisk together eggs, oil, milk, and vanilla extract until well combined.
Add the wet ingredients to the dry ingredients, stirring until just combined to avoid overmixing.
Fold the chopped apples into the batter until evenly distributed.
Pour the batter into the prepared loaf pan. Smooth the top with a spatula.
Bake for 50-60 minutes, or until a toothpick inserted into the center comes out clean. Allow to cool in the pan for 10 minutes, then transfer to a wire rack to cool completely.