Melt 2 cup (2 sticks) of unsalted butter in a microwave-safe dish. Make sure to stir to prevent hot spots, then set aside.
Grate your carrots using a grater, making sure to pack the cup well. Once you have enough, pour the carrots into paper towels and squeeze to release the moisture. Repeat 2-5 times depending on how moist your carrots are.
In a large bowl, add the cooled melted butter. Add 1 cup brown sugar and ½ cup of sugar. Beat with hand or stand mixer on high speed for ~2 minutes or until sugar is evenly distributed and begins to dissolve.
Add in the grated carrots, 1 tsp vanilla, and 2 egg yolks. Mix on medium speed until combined.
Add in the dry ingredients: 3 cups flour, 1 tsp baking soda, 1 tsp salt, 2 tsp ground cinnamon, and heaping ¼ tsp ground nutmeg. Mix on low speed until ingredients are just incorporated and no more streaks appear in the dough.
Divide the dough into 15 portions. I used roughly 2 scoops of my cookie scoop for each cookie.
Once the cheesecake filling are frozen, or at least chilled enough to handle without sticking, stuff each cookie. Take one portion of cookie dough and press a thumb in the middle. Fill with cheesecake filling and cover with cookie dough. Roll into a ball, and place on the baking sheet and after repeating with remaining cookies, return to the freezer for 10-20 minutes.
Preheat the oven to 350°F. Remove cookies from the freezer, and arrange on a parchment paper-lined baking sheet. Make sure to allow cookies room enough to spread, I wouldn't recommend more than 5-6 cookies on a baking sheet.
Bake for 13-15 minutes. Once edges begin to brown slightly, remove from the oven, and allow to sit on the pan for 10-15 minutes before transferring