Preheat oven to 350° Fahrenheit. Grease loaf pan with non-stick cooking spray or line with parchment paper.
In medium bowl, whisk together coconut oil, applesauce, apple cider vinegar, and milk. Set aside.
In separate bowl, whisk together flour, cane sugar, brown sugar, salt, baking powder, baking soda, cinnamon, and nutmeg until thoroughly combined.
Using hand mixer, stand mixer, whisk, or silicone spatula, slowly mix dry ingredients into wet ingredients, stirring constantly until fully combined and thick.
Gently fold in grated zucchini and walnuts (optional).
Pour batter into loaf pan, spreading mixture into all corners. Top with additional chopped walnuts if desired.
Bake bread 45 to 50 minutes. To check doneness, insert toothpick or butter knife into center of bread and remove. If utensil comes out completely clean or with few crumbs, baking is done. Top of bread should be lightly browned. Bread will continue cooking slightly while resting.
Remove bread from oven. Let rest in pan until completely cooled, then remove and slice.