- Preheat oven to 400°F. Coat 12 muffin cups with cooking spray.
- Whisk eggs and brown sugar in a medium bowl until smooth. Whisk in bananas, buttermilk, wheat bran, oil and vanilla.
- Whisk whole-wheat flour, all-purpose flour, 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.
- Stir in chocolate chips, if using. Scoop the batter into the prepared muffin cups (they’ll be quite full). Sprinkle with walnuts, if using.
- Bake the muffins until the tops are golden brown and spring back when touched lightly, 15 to 25 minutes. Let cool in the pan for 5 minutes. Loosen edges and turn muffins out onto a wire rack to cool slightly before serving.