Preheat the oven to 350°F (177°C) and grease a 9×13-inch baking pan.

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

Whisk the oil, granulated sugar, brown sugar, applesauce, eggs, and vanilla extract together in a medium bowl. Pour the wet ingredients into the dry ingredients and whisk until combined. Fold in the shredded zucchini until combined.

Pour and spread batter evenly into the prepared pan. Bake for 45–50 minutes. Baking times vary, so keep an eye on it. Around the 30-minute mark, loosely tent with aluminum foil to prevent over-browning. The cake is done when a toothpick inserted in the center comes out clean.

Remove the cake from the oven and set the pan on a wire rack. Allow to cool completely.

Set out a medium heatproof bowl because you’ll need it at the end of this step. Place the sliced butter in a light-colored skillet or saucepan. Melt the butter over medium heat and stir or whisk constantly. After about 5–7 minutes, the butter will begin browning. Once it’s a light caramel-brown color, remove from heat immediately and pour into the bowl. Cool for 10 minutes, then transfer to the refrigerator to cool completely and solidify.

In a large bowl using a handheld or stand mixer, beat the solidified brown butter on high speed until creamy and smooth. Add the cream cheese and beat again until smooth and combined. Add confectioners’ sugar, vanilla, and salt. Beat on low speed for 30 seconds, then switch to high speed and beat for 2 minutes. Spread the frosting on the cooled cake.

Slice and serve. Cover leftover cake tightly and store in the refrigerator for up to 5 days.