Preheat the oven to 350°F (175°C).
In a bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, and salt.
In a separate bowl, blend the granulated sugar, brown sugar, and vegetable oil until combined.
Add the eggs one at a time, mixing well after each addition, then stir in the vanilla.
Gradually add the dry ingredients to the wet mixture, stirring until just combined. Fold in the grated carrots.
Using a tablespoon or cookie scoop, drop rounded mounds of dough onto a baking sheet lined with parchment paper, spacing them about 2 inches apart.
Bake for 12-15 minutes or until the tops are set and spring back when gently pressed.
Remove from the oven and let the cookies cool on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely.
In a mixing bowl, combine softened cream cheese, powdered sugar, and milk, beating until smooth and creamy.
Once cookies are completely cooled, spread a generous amount of frosting on the flat side of one cookie and sandwich it with another.