Preheat the oven
Preheat the oven to 350°F (175°C).
Toast the wheat bran
Spread the wheat bran in a single layer on a baking sheet. The parchment paper will help with the transfer, but I don’t find it necessary to line the sheet. Toast in the oven for 8 to 10 minutes, stirring halfway through, until nutty aroma is released. Be careful not to brown too much. Let it cool a little on the tray, while you prepare the batter and muffin pans.
Prepare the muffin pan
Increase the oven temperature to 425°F (220°C). Grease the tops of two standard muffin tins lightly with nonstick spray. To encourage airflow and even baking, line every other well with paper liners.
Mix the dry and wet ingredients together
In a medium bowl, whisk the toasted wheat bran, white whole wheat flour, baking powder, baking soda, salt and ground cinnamon.
In another large mixing bowl, mix the light brown sugar, honey and melted butter together with a whisk until smooth. Add the eggs one at a time, scraping well after each addition. Add the applesauce (or banana), then stir in the buttermilk and vanilla extract.
Mix the batter
Using a silicone spatula, fold the dry mixture into the wet ingredients. Gently fold briefly or just until no streaks of flour are visible — be careful not to overmix. Spoon the batter into the prepared muffin liners, filling each nearly to the top.
Bake the muffins
Put the muffin tins into the oven and roast at 425 ˚F (220 ˚C) for 5 minutes. Without opening the door of the oven, lower the temperature to 350°F (175°C) and bake for 10 to 13 minutes more. The muffins are done when a toothpick inserted into the center comes out clean.