Preheat the oven to 350°F (175°C), grease a 9x13-inch baking pan, and lightly squeeze the grated zucchini to remove only the excess surface moisture.
Whisk the flour, baking soda, baking powder, cinnamon, and salt together in a large bowl until evenly combined.
In a separate bowl, whisk the eggs, brown sugar, oil, and vanilla until the mixture becomes smooth and slightly thickened.
Stir the grated zucchini into the wet ingredients, then gradually fold in the dry mixture until no large pockets of flour remain.
Fold in most of the toasted walnuts, reserving a small handful for scattering over the top if desired.
Spread the batter evenly in the prepared pan and bake for 34–38 minutes until the center springs back lightly and a toothpick comes out clean.
Allow the cake to cool completely before dusting with powdered sugar or adding cream cheese frosting and the reserved walnuts.