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Prepare a large cookie sheet with cooking spray, parchment paper or a silicon liner. Set aside.
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Use a food processor or vegetable peeler to finely grate the carrots. Squeeze them out in a paper towel if they are really wet (this usually happens when you food process them). Set the carrots aside.
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In the bowl of an electric mixer, combine the butter, brown sugar and granulated sugar. Beat at medium speed until the mixture is light and creamy. Add in the egg and vanilla and stir to combine.
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In a small bowl, combine the flour, baking soda, cinnamon, nutmeg and salt. Add the dry ingredients to the bowl with the wet ingredients and mix together on low until just combined. Fold in the grated carrots. Cover the dough and refrigerate for 1 hour.
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Preheat the oven to 350° F. Use a 1/3 cup measuring cup to scoop out large balls of dough. Flatten the dough out as best as you can (it will be sticky!) and place a Carrot Cake Oreo in the center of the flattened dough. Fold the dough around the Oreo as best as you can, making sure that the Oreo is completely inside the dough.
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Place no more than 6 cookies on each baking pan (they are large and will spread!). Bake for 10-12 minutes until the cookies are lightly browned around the edges. Allow them to cool completely for about 30 minutes before frosting.
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To make the cinnamon cream cheese frosting, beat together the softened butter and cream cheese. Add in the vanilla, cinnamon and salt. Then, slowly add in the powdered sugar, 1/2 cup at a time, mixing to combine. Beat the frosting until completely smooth, adding more powdered sugar if frosting is too thin. (If the frosting is too thick, add in a tbsp of milk!)
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Transfer the frosting to a piping bag and swirl on the top of each cookie. Then, place another Carrot Cake Oreo on top of each cookie. Enjoy!