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1
Preheat your oven to 350u00b0F Grease or line two standard 12-cup muffin tins, and set them aside.
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2
Make the cupcake batter. In a large bowl, place the flour, xanthan gum, baking soda, baking powder, salt, cinnamon and granulated sugar. Whisk to combine well. Add the brown sugar, and whisk again to combine, working to break up any lumps in the brown sugar. Add the grated carrots and chopped nuts to the dry ingredients, and mix to combine.
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3
In a separate, medium-sized bowl, place the oil, eggs, vinegar and milk, and whisk to combine very well. The mixture should be become pale and frothy. Working quickly, create a well in the center of the bowl of dry ingredients, and pour in the oil and eggs mixture. Mix to combine. The batter will be thick, but soft.
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4
Bake the cupcakes. Fill the wells of the prepared muffin tins about 3/4 of the way full with batter, and shake the pan back and forth to distribute the batter evenly in each well. One at a time, place the muffin tins in the center of the preheated oven and bake until a toothpick inserted in the center of the cupcakes comes out with no more than a few moist crumbs attached (about 20 minutes). Remove the tin from the oven from the oven and allow the cupcakes to cool in the pan for at least 5 minutes before transferring to a wire rack to cool completely. Repeat with the remaining muffin tin.
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5
Make the frosting. While the muffins are cooling, make the frosting. In the bowl of your stand mixer fitted with the paddle attachment (or a large bowl with a handheld mixer), place the cream cheese and butter. Beat on medium speed until light and fluffy. Add the salt, and 2 cups of confectioners' sugar. Mix on low speed until the sugar is absorbed. Then, turn the mixer to high speed and beat until the frosting is thickened and fluffy. Add more confectioners' sugar as necessary to thicken the frosting enough to hold its shape when scooped with a spoon. Spoon or pipe the frosting onto the cooled cupcakes.
