Preheat oven to 400°F. Coat 12 muffin cups with cooking spray or line with paper cups.
Whisk eggs and brown sugar in a medium bowl until smooth. Mash bananas with a fork; in a separate bowl, mix together 1 cup banana, buttermilk, wheat bran, oil, and vanilla. Whisk flours, baking powder, baking soda, cinnamon, and salt in a large bowl.
Make a well in the dry ingredients, add the wet ingredients, and stir with a rubber spatula until just combined. Scoop the batter into the prepared muffin cups (they’ll be quite full).
Bake the muffins 15-25 minutes until the tops are golden brown and spring back when lightly touched. Let cool in the pan for 5 minutes, then transfer to a wire rack to finish cooling.
Serve with a small glass of milk and extra bananas if you have them.