Prepare the baking sheets: Preheat the oven to 350°F (175°C). Line two large baking sheets with parchment paper.

Mix the dry ingredients: In a medium bowl, whisk together the flour, baking powder, baking soda, kosher salt, cinnamon, ginger, and nutmeg.

Cream the butter and sugar: In a large bowl, beat the softened butter and brown sugar on medium speed for 2–3 minutes, until smooth and creamy.

Add the egg and vanilla: Beat in the egg until fully incorporated, then mix in the vanilla. Scrape down the sides of the bowl so the mixture is evenly combined.

Combine the dough: Add the dry ingredients to the butter mixture and mix on low speed just until almost combined. Fold in the finely grated carrots with a rubber spatula. The dough should be soft and slightly sticky but thick enough to scoop.

Chill the dough: Cover the bowl and refrigerate the dough for 30 minutes. Chilling gives the flour time to hydrate and firms the butter, helping the cookies maintain their shape.

Scoop the cookies: Use a 1 1/2-tablespoon cookie scoop to portion the dough onto the prepared baking sheets, leaving about 2 inches (5 cm) between cookies. Gently flatten the tops if the dough forms tall mounds.

Bake: Bake for 10–12 minutes, one sheet at a time, until the edges are set and lightly golden while the centers still look slightly soft. Do not wait for the centers to become completely firm.

Cool: Let the cookies cool on the baking sheet for 5 minutes, then transfer them to a wire rack. Cool completely before frosting.

Make the frosting: Beat the softened cream cheese and butter together until smooth. Add the powdered sugar and vanilla and beat on low speed until incorporated. Add the milk and beat until smooth and spreadable. If the frosting is too thin, refrigerate it for 10–15 minutes.

Frost the cookies: Spread a thin layer of cream cheese frosting over each completely cooled cookie. Let the frosting sit for about 10 minutes before stacking or serving.