Preheat oven to 350°F (175°C). Grease a 9x5-inch loaf pan and line it with parchment paper. Whisk the flour, baking powder, baking soda, salt, cinnamon, and nutmeg in a large bowl.

In a medium bowl, whisk the eggs, granulated sugar, brown sugar, oil, applesauce, and vanilla until smooth.

Pour the wet ingredients into the dry, stirring just until no flour streaks remain. Fold in the diced apple and nuts if using. Scrape the batter into the pan.

Bake 50 to 60 minutes, until a toothpick inserted in the center comes out with a few moist crumbs. Cool in the pan on a wire rack for 15 minutes, then lift out and cool completely.

Slice with a serrated knife and serve.