Preheat the oven to 350°F (175°C), grease a 9x5-inch loaf pan, and lightly squeeze the grated zucchini to remove excess liquid.
Whisk flour, baking soda, cinnamon, and salt in a medium bowl until evenly blended.
In a larger bowl, whisk the eggs, brown sugar, oil, and vanilla until smooth and slightly thickened.
Stir the zucchini and grated carrot into the wet ingredients until the vegetables are evenly distributed.
Add the dry ingredients and fold gently until just combined, stopping as soon as no large flour pockets remain.
Spread the batter into the prepared loaf pan and bake for 50–55 minutes until a toothpick inserted into the center comes out clean.
Cool in the pan for 15 minutes before transferring to a rack, then slice only after the loaf has mostly cooled.