Don't forget that butter is only about 80-85% fat (the rest is mostly water, with a little bit of milk solids), so if you replace butter with oil (which is pure fat) you should probably cut down the volume of oil accordingly.
If it's in the cake, chances are your plan of replacing 1/3 of the butter will probably work. The cake will be heavier, but that isn't necessarily a bad thing. Let us know how it goes.
Olive oil (unless you choose extra light tasting) has a pretty strong flavor that may or may not work with your cake. Most carrot cakes are made with a vegetable oil. If you are referring to a frosting (especially a buttercream), oil probably won't work.
Hmm, could you alter this recipe to make it white chocolate?
http://www.tasteofhome.com/recipes/chocolate-carrot-cake
Or do you mean white chocolate icing with a carrot cake?
http://www.chatelaine.com/recipe/quick-and-easy/carrot-cake-with-white-chocolate-icing/ (the butter is in the icing only).
No the white chocolate is incorporated into the cake. Damn this is a hard one, I am just going to give it a go and maybe substitute 1/3 of the butter with oil
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http://www.tasteofhome.com/recipes/chocolate-carrot-cake
Or do you mean white chocolate icing with a carrot cake?
http://www.chatelaine.com/recipe/quick-and-easy/carrot-cake-with-white-chocolate-icing/ (the butter is in the icing only).