Preheat the oven to 350°F (177°C) and grease a 9×13-inch pan. I always use this glass pan. See recipe Notes for other pan size options.

In a large bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, ginger, nutmeg, and cloves. Set aside.

In a medium bowl, whisk together the oil, brown sugar, applesauce, eggs, vanilla extract, and molasses (if using). Pour the wet ingredients into the dry ingredients and whisk until combined. Some small lumps are OK. Fold in the shredded apple until combined.

Spread the batter evenly into the prepared pan. Bake for 38–40 minutes. Baking times vary, so keep an eye on yours. The cake is done when a toothpick inserted in the center comes out clean with just a few moist crumbs. If you find the top of the cake is browning too quickly in the oven, loosely cover it with aluminum foil (I usually do this around the 25-minute mark).

Remove the cake from the oven and set on a cooling 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. A light-colored interior is crucial for determining when the butter begins to brown. Melt the butter over medium heat and stir or whisk constantly. Once melted, the butter will begin to foam. Continue stirring/whisking, keeping a close eye on it. After about 5–7 minutes, the butter will begin browning and you’ll notice lightly browned specks forming at the bottom of the pan. The butter will have a nutty aroma. The color will gradually deepen, from yellow to golden to golden-brown; once it’s a light caramel-brown color, immediately remove from heat and pour into the bowl, including any brown solids that have formed on the bottom of the pan. Cool for 10 minutes, then transfer to the refrigerator to cool completely and solidify.

In a large bowl using a handheld or stand mixer fitted with a paddle attachment, beat the solidified brown butter on high speed until creamy and smooth. Add the cream cheese and beat again until smooth and combined. Scrape down the sides and bottom of the bowl as needed. Add confectioners’ sugar, vanilla, and salt. Beat on low speed for 30 seconds, then gradually increase to high speed and beat for 2 minutes.

Spread the frosting on the cooled cake. If desired, top with a sprinkle of cinnamon or nutmeg. Refrigerate for 30 minutes before serving. This helps sets the frosting and makes slicing easier.

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