Preheat oven to 350° F. Spray a muffin pan with cooking spray and set it aside.
Wrap the shredded zucchini in a clean kitchen towel or paper towels and squeeze out excess moisture. Set aside.
Place the butter in a large microwave-safe mixing bowl and melt it in the microwave. (Or, you can melt the butter in a pan on the stove.)
To the bowl with the butter, add the mashed banana (or applesauce), honey (or pure maple syrup), vanilla and eggs. Whisk until well combined.
Add the cinnamon, baking power, baking soda and salt. Whisk until very well combined, making sure there are no clumps of baking soda or baking powder in the batter. Add the white whole wheat flour and mix it in using a rubber spatula until just combined. Stir in the zucchini (and chocolate chips or nuts, if using), being careful to not over-mix the batter.
Scoop the batter into the muffin pan, dividing it evenly between the 12 muffin wells. Bake muffins fro 16 to 18 minutes, or until a tester inserted into the center of a muffin comes out clean. Let muffins cool in the pan for 5 minutes and then transfer to a wire rack to finish cooling.