Beat room temperature cream cheese in a bowl until smooth and creamy, about 2 minutes. Gradually mix in granulated sugar and vanilla extract until fully combined. Dollop onto wax paper and freeze for at least 30 minutes.

Finely grate fresh carrots and absorb excess moisture with a towel. In a bowl, sift together flour, baking soda, salt, cinnamon, and nutmeg. Set mixture aside.

In a stand mixer, beat melted butter with brown and granulated sugar on medium speed for about 2 minutes until light and fluffy. Add carrots, vanilla, and egg yolks, mixing until well blended.

On low speed, gradually add dry ingredients to wet mixture until just combined. Avoid overmixing to keep cookies chewy.

Preheat oven to 350°F (175°C). Chill dough in refrigerator for about 15 minutes. Scoop dough, roll into balls, and create a dent in each. Place on lined baking sheet.

Bake for 13-15 minutes until edges are lightly golden. Cookies should be soft in the center. Cool on sheets for a few minutes before transferring to a wire rack.

Beat room temperature butter until creamy, gradually mix in powdered sugar until smooth. Pipe decorations on cooled cookies if desired, and let cookies rest to set frosting.