Break the eggs in a medium bowl, and add brown sugar in it. Stir it together until the mixture becomes smooth. Add bananas, buttermilk, wheat bran, oil and vanilla in the mixture, and whisk it together properly.
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Take a large bowl, and add whole-wheat flour, all purpose flour, baking soda, baking powder, cinnamon and salt. Now whisk all the ingredients together, and then make a well in the mixture. Add the wet mixture into it, and beat them together with a rubber spatula until they are completely mixed. Add chocolate chips if using and stir them properly. Now scoop the batter into the coated muffin cups. Sprinkle the walnuts at the top of it if using.
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Put the muffins in the preheated oven, and bake them for about 15 to 20 minutes until a golden brown top is achieved. Let it cool for some time before taking them out of the cups. Serve as you need.
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