Preheat your oven to 375°F (190°C). Line a 12-cup muffin tin with paper liners or lightly coat with cooking spray.
In a large bowl, whisk together the wheat bran, whole wheat flour, all-purpose flour, baking powder, baking soda, salt, cinnamon, and nutmeg.
In a separate medium bowl, beat the eggs slightly, then add applesauce, honey (or maple syrup), brown sugar, buttermilk, oil, and vanilla extract. Whisk until well combined and smooth.
Make a well in the center of your dry ingredients and pour in the wet mixture. Stir just until combined – about 10-15 strokes with a wooden spoon or spatula.
Gently fold in the diced apples, distributing them evenly throughout the batter.
Divide the batter evenly among the prepared muffin cups, filling each about ¾ full.
Bake for 20-22 minutes, or until a toothpick inserted into the center comes out clean or with just a few moist crumbs. The tops should be golden brown and spring back slightly when touched.
Allow the muffins to cool in the pan for 5 minutes before transferring to a wire rack to cool completely.