Preheat oven to 350 degrees F (175 degrees C). Butter three 9-inch round cake pans. Line bottoms with wax paper; grease wax paper with butter.
Combine cooked carrots, flour, white sugar, walnuts, pineapple, corn oil, eggs, cinnamon, baking soda, 1 tablespoon vanilla extract, and salt in a large bowl.
Pour carrot mixture into a food processor, and blend, working in batches if necessary, until smooth. Divide batter among prepared cake pans.
Bake cakes in the preheated oven until a toothpick inserted into the centers comes out clean, about 35 minutes. Cool in the pans for 5 minutes before inverting onto wire racks to cool completely, about 1 hour.
Beat cream cheese, 1 1/4 cup butter, lemon juice, and 2 teaspoons vanilla extract together in a large bowl until smooth. Gradually beat in confectioners' sugar until frosting is smooth.
Place 1 cake layer on a serving plate and spread with 1/4 of the frosting. Place a second cake layer on top of the first; spread 1/4 of the frosting on top. Place third cake layer on top; frost top and sides with remaining 1/2 of the frosting.