Preheat your oven to 350°F (175°C).
Grease a rectangular baking dish or line it with parchment paper.
In a large bowl, combine the beaten eggs, brown sugar, canola oil, honey, and vanilla extract until smooth.
Fold in the grated carrot, zucchini, and chopped walnuts into the wet ingredients until well combined.
In another bowl, whisk together flour, baking powder, ground ginger, and baking soda until evenly mixed. Gradually add this dry mixture to the wet ingredients, stirring gently until just combined.
Pour the batter into your prepared baking dish and spread it evenly. Bake for about 25 minutes or until a toothpick inserted in the center comes out clean; let cool completely before frosting.