Preheat oven to 350 degrees F (175 degrees C). Spray a 9x13-inch baking dish with cooking spray.
Sift flour, baking soda, cinnamon, and 1/2 teaspoon salt together in a bowl. Lightly beat eggs in a bowl; add white sugar, buttermilk, applesauce, oil, and 2 teaspoons vanilla extract and mix well. Stir buttermilk mixture, carrots, pecans, and drained pineapple into flour mixture just until mixed; pour into prepared dish.
Bake in the preheated oven until a toothpick inserted in the center of the cake comes out clean, 45 to 50 minutes. Cool cake in baking dish on a wire rack, about 1 hour. Cover dish tightly with plastic wrap; refrigerate 8 hours to overnight.
Beat Neufchatel cheese and 1 tablespoon reserved pineapple juice together in a bowl with an electric mixer until smooth; add confectioners' sugar, 1 teaspoon vanilla extract, and 1 pinch salt and beat until frosting is smooth. Spread frosting on cake.
