In a large bowl, cream together the softened butter and brown sugar with a hand mixer or stand mixer until creamy. Beat in the eggs and vanilla extract until combined.

In a separate bowl, quickly whisk together the flour, baking soda, baking powder, cinnamon, ground ginger, ground nutmeg, and salt. Sift these dry ingredients into the wet ingredients.

Fold together with a silicone spatula. When the batter is almost combined, fold in the grated carrots until just combined.

With a 1" cookie scooper, scoop out small rounds of dough and place on lined baking sheet, spacing them about 2 inches apart. If the dough is too sticky, let it chill in the fridge for 15-30 minutes.

Bake the cookies for about 15 minutes. Remove from oven and let cool completely before filling.

In a large bowl, beat together the softened butter and cream cheese with a hand mixer or stand mixer until pale and creamy, about 3 minutes. 

Add in the cinnamon, vanilla extract, powdered sugar, and milk. Continue beating until it's fluffy and creamy, about 3 minutes.

Transfer cream cheese frosting to a piping bag fitted with a plain circular or star tip. Pipe frosting onto one carrot cake cookie and sandwich with another similarly sized cookie.