Step 1: Preheat your oven to 350°F (175°C) and line two large baking sheets with parchment paper or silicone baking mats.
Step 2: In a medium bowl, whisk together the flour, baking powder, baking soda, cinnamon, ginger, nutmeg, and salt until well combined.
Step 3: In a large bowl using a hand mixer or stand mixer, cream together 1/2 cup softened butter, brown sugar, and granulated sugar until light and fluffy.
Step 4: Beat in the egg and vanilla extract until smooth, then stir in the finely grated carrots by hand.
Step 5: Gradually add the dry ingredients to the wet ingredients, mixing until just combined. Do not overmix.
Step 6: Use a cookie scoop to drop rounded tablespoons of dough onto the prepared baking sheets, spaced 2 inches apart.
Step 7: Bake for 10-12 minutes or until the edges are set and the centers spring back when lightly touched.
Step 8: Allow the cookies to cool on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely.
Step 9: For the filling, beat the cream cheese and 1/4 cup butter together until smooth. Gradually add powdered sugar and vanilla, beating until fluffy.
Step 10: Once the cookies are cold, pipe or spoon a generous amount of frosting onto the flat side of half the cookies. Top with the remaining cookies to create sandwiches.