Preheat oven to 425° 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 and then pour it into your mixing bowl.)
To the bowl with the melted butter, add the applesauce (or mashed banana), honey (or pure maple syrup), eggs and vanilla. 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 flour and mix it in using a rubber spatula until almost 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 greased muffin pan, dividing it evenly between the 12 muffin wells. Tap the muffin pan on the countertop a few times (I place a folded kitchen towel underneath to protect the counter) to even out the tops of the muffins.
Bake muffins for 5 minutes at 425°F, then reduce the oven temperature to 350°F and bake for 11-13 more minutes, or until a toothpick 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.