To begin, preheat your oven to 350 degrees F (175 degrees C) and grease a 9x13-inch baking dish with butter. This will ensure that your cake won't stick to the dish.
Next, combine grated carrots, brown sugar, white sugar, and lemon zest in a bowl. Toss to coat and allow the natural juices to come out. This will make your cake moist and flavorful. Set the mixture aside.
In a separate bowl, mix your flour, baking powder, cinnamon, and nutmeg together. This will ensure that your cake is evenly mixed and baked.
In a large bowl, whisk together your eggs, applesauce, oil, and vanilla extract. Add the flour mixture 1/4 cup at a time, mixing until just combined. This will ensure that your cake is light and fluffy.
Pour the batter into the prepared baking dish and bake in the preheated oven until a toothpick inserted into the center comes out clean, about 45 minutes. This will ensure that your cake is fully cooked and delicious.
Cool the cake on a wire rack for 5 minutes. Run a table knife around the edges of the pan to loosen. Invert the cake carefully onto a serving plate or cooling rack. Let it cool completely before serving.
