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Pre-heat the oven to 425°Butter the inside of 12 muffin pan cups and set the pan aside.
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Mash the bananas with a fork in a large bowl. Add the eggs, applesauce, melted butter and sugar and mix until well combined.
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In a separate bowl, whisk together the all-purpose flour, whole wheat flour, baking soda, baking powder and salt. Add the dry ingredients to the wet ingredients and mix by hand, just until the dry ingredients are incorporated into the wet. Do not over-mix the batter. Gently fold in the chocolate chips.
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Fill the muffin cups ¾ of the way to the rim and transfer the pan to the middle rack in the oven.
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Bake at 425°F for 8 minutes. Then turn the oven temperature down (do not open the oven door at this pointo 350°F and bake for 10 to 12 minutes, until a toothpick inserted into the center of a muffin comes out clean.
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Transfer the muffin pan to a cooling rack and let it cool for 3 minutes. Then remove the muffins from the pan and cool completely on the cooling rack. (Run a butter knife around the edges if necessary, to loosen the muffins from the pan.)