Preheat oven to 350 degrees F (175 degrees C). Grease a 9x13-inch cake pan.
Whisk honey, grapeseed oil, eggs, and egg whites together in a bowl.
Mix whole wheat flour, cinnamon, baking soda, baking powder, salt, nutmeg, ginger, and allspice together in a large bowl. Make a well in the center; pour in honey mixture. Mix gently until batter is just combined.
Fold carrot pulp, apple, and walnuts into the batter. Pour batter into the prepared cake pan.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, 35 to 40 minutes. Set pan on a wire rack and let cake cool completely, about 30 minutes.
Combine Neufchatel cheese, confectioners' sugar, vanilla extract, and lemon zest in a bowl; beat with an electric mixer until frosting is smooth. Spread frosting over cooled cake.
