Preheat your oven to 350°F (177°C). Grease a 9×13-inch baking pan generously to prevent the cake from sticking.

In a large bowl, whisk together the all-purpose flour, baking powder, baking soda, salt, cinnamon, and nutmeg until well combined. Set this bowl aside for later use.

In a separate medium bowl, whisk the vegetable oil, brown sugar, white sugar, applesauce, eggs, and 1 tablespoon of vanilla extract together until smooth and homogeneous. Pour the wet ingredients into the bowl containing the dry ingredients. Whisk the mixture until just combined—do not overmix. Finally, fold in the shredded zucchini until evenly distributed throughout the batter. For extra moisture, I don't squeeze the zucchini dry unless it seems unusually wet.

Pour and spread the batter evenly into your prepared 9×13-inch pan. Bake for 45–50 minutes, keeping an eye on the cake as ovens may vary. About 30 minutes into baking, tent the cake loosely with aluminum foil to prevent over-browning. The cake is finished when a toothpick inserted in the center comes out clean. Once baked, place the pan on a wire rack and allow the cake to cool completely before frosting. I recommend letting the cake cool in the pan so it remains moist.

While the cake cools, set out a medium heatproof bowl. Place the sliced unsalted butter in a light-colored skillet or saucepan over medium heat. Melt the butter, stirring or whisking constantly as it foams. After 5–7 minutes, it will begin to brown, developing a nutty aroma and brown bits at the bottom. Once it reaches a light caramel-brown color, immediately remove from the heat and pour all contents (including brown solids) into the prepared bowl. Cool for 10 minutes, then chill in the refrigerator until completely solidified. I love using browned butter for extra depth of flavor in the frosting.

In a large bowl, use a handheld or stand mixer to beat the solidified browned butter (from Step 5) on high speed until creamy and smooth. Add the softened cream cheese and beat again until fully combined. Scrape down the sides and bottom of the bowl as needed. Gradually add the powdered sugar, 1 teaspoon vanilla extract, and a pinch of salt. Beat on low for 30 seconds, then increase to high speed and beat for another 2 minutes, creating a light and fluffy frosting.

Spread the cream cheese frosting (from Step 6) evenly over the completely cooled cake (from Step 4). Slice into pieces and serve. For a perfectly smooth finish, I like to use an offset spatula to spread the frosting. Store any leftovers tightly covered in the refrigerator for up to 5 days.