Preheat your oven to 350°F. Grease a 9x5-inch loaf pan or line it with parchment paper, leaving overhang on the long sides for easy removal.
Grate the zucchini and carrots using the large holes of a box grater. Place the grated zucchini in a colander, sprinkle with a pinch of salt, and let it sit for 5 minutes. Then transfer to a clean kitchen towel or paper towels and squeeze out as much liquid as possible. This step is non-negotiable — it prevents a soggy loaf.
In a large bowl, whisk together the flour, cinnamon, nutmeg, salt, baking soda, and baking powder.
In a separate bowl (or the same bowl if you push the dry ingredients aside), whisk together the brown sugar, granulated sugar, oil, applesauce, eggs, and vanilla until smooth.
Pour the wet ingredients into the dry ingredients and fold gently with a rubber spatula until just combined. A few streaks of flour are fine — overmixing makes the bread tough. Fold in the grated carrots, squeezed zucchini, and nuts if using.
Pour the batter into the prepared pan and smooth the top. Bake for 50 to 60 minutes, until a toothpick inserted in the center comes out clean or with just a few moist crumbs. The top should be golden brown and spring back when gently pressed. If browning too quickly around 40 minutes, tent loosely with foil.
Let the bread cool in the pan for 10 minutes, then turn it out onto a wire rack to cool completely. Resist slicing warm — it sets as it cools and will hold together much better.