I didn't discover Cream Cheese Frosting until I was an adult and Mum and I made our first carrot cake. She came up with a recipe using baby food because there was a ready supply when my son was eating them. Simply the most moist carrot cake I've ever eaten. If I'm making the cake for our small family of 3, I usually cut the cake in half and freeze half and frost the rest for immediate consumption :) This cake freezes well for about a month, wrapped first in clear wrap, then foil then I put it in a plastic container to protect it. If you are freezing the cake, DO NOT frost it first.
Mix eggs, sugar and oil together. Sift together and add dry ingredients to wet mix, then vanilla. Fold in carrots.
Bake in preheated 350° oven in a lightly greased and floured 9 x 13 pan for 40 to 45 minutes.
Cool completely before frosting with your favourite such as cream cheese, butter cream or simple french vanilla frostings. I use canned frostings for the very sporadic times I bake cakes. Mum would make her icing from scratch.