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Preheat the oven to 350ºF.
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Line a muffin tin with baking liners.
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In a medium sized bowl, combine flour, cinnamon, baking soda, baking powder, nutmeg and salt.
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Set the dry ingredients aside.
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In a stand mixer, beat the eggs and sugar until frothy. (This step could also be done with a hand mixer or just good old fashioned arm power, but a stand mixer makes it simple and quick!)
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Add the applesauce, oil, 1 teaspoon of vanilla extract, and carrot to the bowl of the stand mixer. Combine well.
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Add the flour mixture to the wet mixture and combine until moistened.
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Pour the batter into the lined muffin tin.
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Bake the muffins for 18-20 minutes, checking doneness with a toothpick (they are done when the toothpick comes out clean).
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Remove the muffins from the tin and cool on a wire rack.
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Make the frosting by beating the cream cheese and butter in the stand mixer.
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Add the confectioner's sugar and remaining 1 teaspoon of vanilla extract and beat until thick and creamy.
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Use milk to thin the cream cheese frosting until the desired consistency.
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Frost the muffins and garnish as desired!