Preheat oven to 350 degrees F (175 degrees C).

Grease a 9x5-inch loaf pan.

In a large bowl, combine whole wheat flour, brown sugar, flaxseed meal, oats, white sugar, baking powder, baking soda, and salt.

In another bowl, combine eggs, melted butter, 1/3 cup plus 1 tablespoon buttermilk, and vanilla extract.

Stir the wet ingredients into the dry ingredients until just combined.

Add mashed bananas and mix until the batter is well-combined.

Pour the batter into the prepared loaf pan.

Bake in the preheated oven for 50 minutes to 1 hour, or until a toothpick inserted into the center comes out clean.

Cool in the pan for 5 minutes.

Invert the bread onto a wire rack and let cool for about 25 minutes before slicing and serving.