Yes, you can certainly use melted butter in place of oil. Butter at room temperature is called for in cakes where emulsion of fats and sugars is important. In cakes such as carrot cake, emulsion is not important because it is a dense cake by virtue of its very weighty ingredients. It doesn't require the additional leavening that results from highly whipped butter and sugar, and in fact would be destabilized if it occurred.
Hi-- I'm not an oil-in-baking expert, but I know that butter has milk solids and water in it, whereas whichever oil the recipe calls for would not have those. I would use clarified butter, aka ghee. That is all oils. Depends how much it calls for and what the purpose of it is, I suppose.
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