Preheat the oven to 350 degrees F (175 degrees C). Grease a 9x13-inch baking dish with butter.

Combine grated carrots, brown sugar sugar, white sugar, and lemon zest in a bowl. Toss to coat and allow the natural juices to come out. Set aside.

Mix flour, baking powder, cinnamon, and nutmeg together in a separate bowl.

Whisk eggs, applesauce, oil, and vanilla extract together in a large bowl. Add flour mixture 1/4 cup at a time, mixing until just combined. Fold in carrot-sugar mixture with all the juice that has accumulated. Add food coloring. Pour batter into the prepared baking dish.

Bake in the preheated oven until a toothpick inserted into the center comes out clean, about 45 minutes.

Cool cake on a wire rack for 5 minutes. Run a table knife around the edges of the pan to loosen. Invert cake carefully onto a serving plate or cooling rack. Let cool completely.