Preheat oven to 350 degrees F.
Spray a 9x5-inch loaf pan well with cooking spray and lightly flour the pan.
Wrap the shredded zucchini in a clean kitchen towel and squeeze out the excess moisture. Set aside.
Place the butter (or coconut oil) in a large bowl and melt in the microwave. Let cool slightly if very hot. Add the applesauce, honey or maple syrup, milk and vanilla. Whisk until well combined. Whisk in the eggs.
Sprinkle on the cinnamon, nutmeg, baking powder, baking soda and salt. Whisk until well combined.
Add the white whole wheat flour and mix with a rubber spatula until almost combined. Gently fold in the zucchini and nuts (if using), being careful to not over-mix.
Transfer the batter to the prepared pan. Bake for 55 minutes to 1 hour, until a tester inserted into the center of the bread comes out clean. Let cool in the pan for 20 minutes and then remove to a wire rack to cool completely.
To store zucchini bread, wrap airtight and store at room temperature for up to 3 days, in the refrigerator for up to 5 days or in the freezer for up to 3 months.