Preheat your oven to 350°F (175°C) and grease a 9x13-inch baking pan.
In a mixing bowl, whisk together the flour, baking powder, baking soda, cinnamon, and salt. Set aside.
In a separate large bowl, combine the vegetable oil, applesauce, granulated sugar, brown sugar, eggs, and vanilla extract. Mix until well blended.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined.
Fold in the grated zucchini and chopped walnuts or pecans, if using, until evenly incorporated.
Pour the batter into the prepared baking pan and smooth the top with a spatula.
Bake in the preheated oven for 25-30 minutes or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and let it cool completely in the pan on a wire rack.
While the cake cools, prepare the cream cheese frosting. In a bowl, beat together the softened cream cheese and butter until smooth.
Gradually add the powdered sugar and mix until well combined. Add vanilla extract and milk, adjusting for desired consistency.
Once the cake is completely cool, spread the cream cheese frosting evenly over the top.
Slice into pieces and serve.