Brown the butter in a small saucepan over medium to medium-low heat. Cook until the milk solids are golden brown and the butter smells toasted and nutty. Transfer it to a bowl and allow it to cool slightly.
Grate the zucchini with a box grater. Place it on paper towels and gently blot the surface to remove a little excess moisture. Do not squeeze it dry.
In a large bowl, combine the eggs, applesauce, browned butter, granulated sugar, brown sugar, and vanilla.
In a separate bowl, combine the flour, baking powder, baking soda, salt, cinnamon, ginger, and nutmeg.
Add the dry ingredients to the wet ingredients and fold until almost combined.
Add the grated zucchini and gently fold just until no dry streaks of flour remain.
For mini loaves, divide the batter evenly between four 6 x 3.5 x 2-inch aluminum loaf pans.
Bake until the centers are set and a toothpick comes out clean or with a few moist crumbs, about 25 to 35 minutes for mini loaves. Begin checking at 25 minutes.
Let the loaves cool in their pans for 15 to 20 minutes before transferring them to a cooling rack.
Cool completely before wrapping, storing, or packaging.