Preheat oven to 350° Fahrenheit. Grease loaf pan or line with parchment paper.
In large bowl, whisk together gluten free flour, brown sugar, white sugar, baking powder, salt, cinnamon, and nutmeg. Combine well, then set aside.
In medium bowl, whisk together eggs, coconut oil, applesauce, milk, and vanilla extract. Combine thoroughly.
Slowly pour wet ingredients into dry ingredients. Mix until thoroughly combined.
Gently fold zucchini and walnuts (optional) into batter mixture.
Transfer mixture from bowl to greased (or lined) loaf pan. Top with additional chopped walnuts, if desired. Bake 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 baking slightly while resting.
Remove pan from oven and let bread rest in pan to cool. Cool completely before removing and slicing.