Beat room-temperature cream cheese with granulated sugar and vanilla extract until creamy and smooth, about 2 minutes. Scoop into dollops and freeze until solid, approximately 30 minutes.

Grate fresh carrots finely and pat dry. Sift together flour, baking soda, salt, ground cinnamon, and ground nutmeg in a mixing bowl.

In a large mixing bowl, combine melted butter, light brown sugar, and granulated sugar. Blend on medium speed for 2 minutes until fluffy. Incorporate grated carrots, vanilla extract, and egg yolks until well combined.

Gradually add the sifted dry mixture to the wet ingredients, mixing on low speed just until the flour disappears and the dough comes together.

Cover the bowl with plastic wrap and let it rest in the refrigerator for about 10 minutes. Use a cookie scoop to portion the dough into balls, create a well in each, insert the frozen cream cheese filling, and seal the dough.

Preheat oven to 350°F (175°C). Place stuffed cookie dough balls on parchment-lined baking sheets and bake for 13-15 minutes or until edges are golden.

Remove cookies from the oven and allow to cool on the baking sheet for 10-15 minutes before transferring to a wire rack.