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Preheat the oven to 350F.
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Combine flours, sugar, cinnamon, baking soda, baking powder, and salt in a medium bowl.
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In a separate bowl, combine the butter and carrot.
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Slowly add the flour mixture to the butter/carrot mixture, stirring until just combined.
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Add the eggs and stir to combine a few times, don't over-mix.
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Scoop the batter into muffin liners (I like to fill mine up to the top or you can do 2/3 full).
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Bake 13-16 minutes, checking at 13 minutes for doneness. The tops should be golden brown. Do the toothpick test and add more minutes if necessary as oven temperatures can vary.
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Remove from the oven and let cool on a rack.
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For the frosting, blend the cream cheese, butter and vanilla until smooth.
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Slowly beat in the powdered sugar until completely smooth. Transfer to a plastic bag for piping.
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Frost the cupcakes once they have cooled completely.