Preheat oven to 350°. Melt the coconut oil, then set it aside and allow it to come to room temperature before using.
Next, stir the milk, applesauce, vanilla, and apple cider vinegar together in a large mixing bowl, then add the cooled coconut oil.
In a separate bowl, stir the flour, cane sugar, brown sugar, baking powder, baking soda, cinnamon, and nutmeg together. Then, pour the dry ingredients into the wet and use a rubber spatula to fold them together. Last, stir in the zucchini and walnuts. NOTE: There's no need to squeeze the moisture out of the zucchini -- you can just shred it directly into the bowl.
Grease a muffin tin with nonstick spray, then divide the batter evenly.
Bake the muffins for 20-25 minutes, or until a toothpick comes out clean. Leave the muffins in the tin for 5 minutes, then carefully remove them and transfer to a cooling rack. Enjoy!