Preheat oven to 350 degrees F (175 degrees C). Grease a loaf pan.
Whisk flour, baking soda, baking powder, cinnamon, nutmeg, and salt together in a bowl. Mix banana, white sugar, brown sugar, egg white, milk, canola oil, and vanilla extract together in a separate bowl; stir into flour mixture just until batter is moistened. Fold sesame seeds and walnuts into batter; pour into prepared loaf pan.
Bake in the preheated oven until a toothpick inserted in the center comes out clean, 35 to 40 minutes. Cool in pan before transferring to a wire rack to cool completely.