Whisk the flour, baking powder, baking soda, salt, cinnamon, and nutmeg together in a large bowl. Set aside.

In a separate medium bowl, whisk together the oil, granulated sugar, brown sugar, applesauce, eggs, and vanilla extract.

Add the wet ingredients to the dry ingredients and whisk until combined. Gently fold in the shredded zucchini until evenly incorporated.

Pour the batter into the prepared pan, spreading it evenly. Bake for 45–50 minutes, or until a toothpick inserted in the center comes out clean. Tent with foil around the 30-minute mark to prevent over-browning.

Allow the cake to cool completely on a wire rack.

To make the frosting, melt the butter in a light-colored skillet over medium heat. Stir constantly until the butter foams, then begins to brown with a nutty aroma. Pour into a heatproof bowl, scraping up any browned solids, and let it cool for 10 minutes. Transfer to the refrigerator to solidify.

Once the brown butter has solidified, beat it in a large bowl with a mixer until smooth. Add the cream cheese and continue beating until fully combined and creamy. Add confectioners’ sugar, vanilla, and salt. Beat on low for 30 seconds, then on high for 2 minutes.

Spread the frosting evenly over the cooled cake. Slice and serve.