Preheat the oven to 350°F. Line a baking sheet with parchment paper.
In a large mixing bowl, beat the butter, brown sugar, and granulated sugar until light and creamy, about 2 to 3 minutes.
Add the egg and vanilla extract and mix until fully combined.
Mix in the finely grated carrots.
In a separate bowl, whisk together the flour, cornstarch, baking soda, baking powder, salt, cinnamon, and nutmeg.
Add the dry ingredients to the wet ingredients and mix just until combined. Do not overmix.
Scoop the dough into 12 large portions, about 1/4 cup each, and place them on the prepared baking sheet with space between each cookie.
Gently shape each dough mound so it stays tall rather than flattening it too much.
Bake for 11 to 13 minutes, until the edges are set and the centers still look slightly soft.
Let the cookies cool on the baking sheet for 5 minutes, then transfer them to a wire rack and cool completely.
For the frosting, beat the cream cheese and butter until smooth and fluffy.
Add the powdered sugar and vanilla extract and beat until thick and creamy.
Spread or pipe the frosting over the completely cooled cookies.