Preheat oven to 350°F (175°C). Grease an 8x8-inch baking pan.
In a large bowl, whisk together flour, baking soda, baking powder, salt, cinnamon, and nutmeg.
In another bowl, whisk together sugars, oil, eggs, and vanilla until smooth.
Fold in the grated carrot, zucchini, and walnuts.
Add wet ingredients to dry ingredients and stir just until combined.
Pour into prepared pan and smooth the top.
Bake for 35 minutes or until a toothpick inserted in the center comes out clean.
Let cake cool completely in the pan before frosting.
For frosting: beat cream cheese and butter until smooth, then mix in powdered sugar and vanilla until fluffy. Spread on cooled cake and top with chopped walnuts if desired.