Preheat your oven to 350°F (175°C). Grease and flour a 9x13-inch baking pan or line with parchment paper.
In a medium bowl, whisk together flour, baking powder, baking soda, cinnamon, and salt. Set aside.
In a large mixing bowl, beat eggs, granulated sugar, and brown sugar until light and fluffy.
Add oil, applesauce, and vanilla extract. Mix until smooth.
Gradually stir in the dry ingredients until just combined.
Fold in the grated zucchini and nuts (if using). Do not overmix.
Pour the batter into the prepared pan, spreading it evenly.
Bake for 35–40 minutes, or until a toothpick inserted into the center comes out clean.
Allow the cake to cool completely in the pan.
For the frosting: Beat cream cheese and butter until smooth. Add powdered sugar gradually, then mix in vanilla extract until creamy.
Spread frosting over the cooled cake. Slice and serve.