Preheat an oven to 350 degrees F and grease an 8x4 loaf pan with nonstick baking spray.
In a large bowl, whisk together the melted butter and light brown sugar.
Next, add the egg, applesauce, and vanilla. Whisk everything to combine.
In a small bowl, whisk together the flour, cinnamon, baking powder, baking soda, and salt. Add the wet ingredients to the dry ones and mix just until no large lumps of flour remain. Do not overmix.
Wrap the shredded zucchini in paper towels (or a lint-free dish towel) and squeeze out any excess moisture. Add the zucchini to the batter and mix just until it is combined.
Transfer the batter to the prepared loaf pan and bake it for 50 to 55 minutes, or until a toothpick inserted into the center comes out clean.
Allow the zucchini bread to cool for 10 minutes in the pan and then remove it to a wire rack to cool completely.