Prepare the oven and pan. Grease and flour a 9-by-13-inch baking pan or spray it with baking spray. Alternatively, line the greased pan with parchment paper.
Combine dry ingredients. Combine the flour, baking soda, baking powder, cinnamon, and nutmeg in a medium bowl and whisk to blend.
Combine wet ingredients and sugars. In a large bowl, using a whisk or electric mixer, combine the eggs, oil, granulated and brown sugars, and 1 1/2 teaspoons of vanilla. Whisk or beat until smooth.
Finish the batter. Add the flour mixture to the wet mixture and stir until well blended. Fold in the zucchini, carrots, and chopped nuts (if using).
Bake the cake. Spread the cake batter in the prepared pan and bake for 45 minutes or until a toothpick inserted into the cake comes out clean. Cool in the pan on a rack. Cool completely before frosting.
Make the frosting. Combine the butter and cream cheese. Combine the cream cheese and butter in a mixing bowl and beat until smooth. Add the confectioners' sugar. With the mixer on low, gradually mix in the confectioners' sugar, then add the vanilla. Increase the speed to medium and continue beating until the mixture is fluffy. Add more confectioners' sugar, as needed for spreading.
Finish and enjoy! Frost the cooled cake.
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