Preheat oven to 350 degrees. Grease a 9x5-inch loaf pan. Whisk together the flour, baking soda, baking powder, salt, cinnamon, ginger, and ground nutmeg in a medium bowl.
In a large bowl, stir together the eggs, vegetable oil, grated apple, grated carrots, brown sugar, vanilla extract, and chopped pecans.
Stir the flour mixture into the carrot mixture just until combined. Pour into prepared pan.
Bake for 50-60 minutes or until a toothpick inserted in the center comes out clean. Do not overbake.
Cool the bread in the pan on a wire rack for 5-7 minutes before carefully inverting it onto the wire rack and gently rolling it over. Cool for 2-3 hours before slicing.