I have always loved carrot cake and was inspired to make it when my brother told me about his recent success with a batch. In order to make it gluten-free and keep it delicious, I culled ideas from these 3 different recipes:
1,
2, and
3. I was worried that it wouldn't turn out well, especially when I used olive oil since that's all I had, but the cake turned out to be very moist and delicious. The batter was enough to make one cake plus 6 muffins. What I did was bake the cake the night before serving it, then refrigerate it over night and the following day make the icing, ice the cake and then serve it cool. It turned out excellent and all my friends were pleased with it. It was one of the more involved things I've made: I'd say about 30 min of prep, 1 hr of cooking for the cake, with another 10 min for the icing. It makes for a good 50th recipe, for sure.
Prepare Cake:
-Grate 3 cups worth of carrots (this was about 4-5 carrots, pending on the size)


-In a large bowl, mix 2 cups gluten-free flour, 1 cup brown sugar, 1 cup white sugar, and 1 tsp baking powder
-Beat 3 large eggs in a bowl
-Chop about a handful of dried dates into small pieces

-Mix the carrots, dates, and eggs with the flour, adding 1 cup of cooking oil (I used 1/4 canola, 3/4th olive)
-Add 1 tbsp of vanilla essence and 1 tbsp ground cinnamon

-Pour into lightly greased cake/loaf/muffin pan, leaving room at the top since it rises a significant amount

Cook:
-Preheat oven to 375F/180C and bake for about 50 minutes for the cake, 30 minutes for the muffins
-Be sure to check with a toothpick! The outside should be firm yet depressible and the inside should be moist but not wet

-Let cool on a drying rack and refrigerate if not eating soon

Prepare Icing:
-Let 1/2-1 cup butter and 1 8oz/225g package of cream cheese soften

-Chop/crush 1/2 cup pecans and 1/2 cup walnuts
-Mix the butter, cream cheese, nuts, 1 tsp vanilla, and about a cup of icing sugar (if not sweet enough, add more)

-Spread atop cake

Tips:
-This cake was very flavorful as it was; other things you might want to add would be nutmeg, cloves, ginger, or orange zest. Also, you could substitute/add to the dates with raisins, craisins, dried apricots, candied ginger, etc.
-The batter was quite runny and pour-able (like beaten eggs). That being said, if it's too runny, add a bit more flour
-With the muffins, you can slice in half, put icing in the middle, and piece back together
-I was worried about the olive oil as the batter had a distinctive olive-oily taste, but there was none after cooking. Still, in the future, I'd maybe use 1/2 olive oil and 1/2 something else. Alternatively, I may use 1/2 oil and 1/2 applesauce for a healthier and still-moister cake
-Some recipes I've seen have called for 1 cup of drained pineapple chunks to be used instead of1 cup of carrots - I did not do this but I bet it would contribute well to the moistness as well as add a bit of tart to the cake. Also, I think that maybe doing half carrot and half zucchini would work nicely
-For the icing, what I did was place the butter and cream cheese in the mixing bowl and set that atop the oven while baking something - they slowly melted and made the blending a lot easier.