Start by preparing the carrots. Grate the carrots using the finer side of your box grater. Measure out 2oz (56g), which is roughly ½ cup lightly packed. Spread the shredded carrots onto a plate in an even layer, then place a paper towel on top and gently press down until the towel absorbs the moisture. Repeat this process two more times with clean paper towels. Removing excess moisture from the carrots helps prevent adding extra water to the cookie dough.
Next, prepare the cheesecake filling. In a medium bowl, combine the cream cheese, sugar, and vanilla. Beat with a hand mixer on high speed until the mixture is creamy. Line a baking sheet with waxed paper and spoon teaspoons of the mixture into individual dollops. Place the tray in the freezer for 30 minutes to allow the cheesecake balls to solidify.
For the cookie dough, combine softened butter, brown sugar, and granulated sugar in a large bowl. Beat with an electric mixer on medium-high speed for about 2 minutes until the mixture is pale and fluffy. Add the egg and vanilla, beating until well incorporated, then mix in the blotted, grated carrots.
In a separate medium bowl, whisk together the flour, spices, and baking soda. Add this dry mixture to the butter mixture and gently fold it in until just combined.
Preheat the oven to 350°F and line two large cookie sheets with parchment paper.
Using a 1 ½-oz cookie scoop, portion out the cookie dough. Place each dough ball in your hand, press a frozen cream cheese ball into the center, and wrap the dough around the cheesecake filling. Roll the dough between your hands to smooth it out. Arrange the cookie balls on the prepared baking sheets, spacing them 2-3 inches apart. Bake for 12-14 minutes, or until the cookies are evenly golden and puffed.
Allow the cookies to sit on the baking sheet for 3-5 minutes before transferring them to a wire rack to cool completely. If storing for longer than 2 days, keep them in an airtight container in the fridge.