Carrot cake is such a classic. My favorite carrot cake recipe, from the Silver Palate, uses pureed carrots which are first boiled. I am convinced the cooking concentrates the sweet carrot flavor because it consistently delivers a cake far superior to those with fresh shredded carrots.
I have found this to be a perfect method for using left-over carrots, puree them and make carrot cake. Lately I have started using brown butter instead of vegetable oil. The resulting cake is so tender with deep, rich flavor.
This last time I topped it off with a brown butter icing: brown butter, sea salt, vanilla, and confectioner's sugar. Dare I say it's better than cream cheese? I haven't decided yet. But it definitely complements the cake well. The salt brings out the nuttiness of the brown butter, and tames the sweetness.
I believe another taste is in order before deciding it beats cream cheese.