Preheat the oven to 350 degrees Fahrenheit (177 degrees Celsius).
Make flaxseed eggs by mixing the 2 tablespoons of ground flaxseed meal with the 5 tablespoons of warm water in a small bowl. Then set the mixture aside for 7-10 minutes to firm up into a gel consistency.
Prepare a 9x5 inch loaf pan by greasing it, dusting with flour, and then lining with parchment paper. The parchment paper is worth the extra step to ensure your bread doesn’t stick to the bottom.
Peel, core, and dice the apples into equal bite size pieces. Set aside.
In a large bowl, sift together the flour, cinnamon, nutmeg, baking soda, baking powder, and salt. Set aside.
In another bowl, beat the oil, flax eggs, sugar, and brown sugar with a whisk until well combined.
Add the vanilla, apple cider vinegar, and almond milk to the flax egg, oil and sugar mixture. Again beat to combine.
Make a well in the center of the dry ingredients and pour in the wet mixture. Use a rubber spatula to gently fold everything together just until combined and there are no large lumps of flour. The batter will be thick. Be careful not to overmix.
Fold in the diced apples and then pour the batter into the prepared loaf pan.
Bake at 350 degrees Fahrenheit (177 degrees Celsius) for 55 to 65 minutes until firm and golden brown. A cake tester should come out clean when the vegan apple bread is done.
Allow the bread to cool on a cooling rack in the pan for 15 minutes. Then turn out from the loaf pan and continue to cool in the cooling rack.