Heat oven to 350°F. Line two baking sheets with parchment.
Whisk flour, baking soda, salt, ginger, cinnamon, and cloves in a bowl.
Cream softened butter with brown sugar and 2 tablespoons granulated sugar until light and fluffy.
Beat in molasses, beaten egg, and vanilla until smooth, scraping the bowl as needed.
Add dry ingredients in two additions. Mix just until a soft dough forms.
Scoop 1-tablespoon portions. Roll each ball in the remaining granulated sugar.
Arrange on sheets, spacing 2 inches apart. Bake 9–11 minutes until edges set and tops crinkle.
Cool 5 minutes on sheets, then move cookies to a rack to cool completely.
Beat cream cheese and butter until perfectly smooth. Gradually mix in powdered sugar.
Blend in vanilla, ground ginger, and salt. Add milk to reach a creamy, spreadable frosting.
Pair similar-sized cookies. Pipe or spread frosting on one, then cap with its partner.
Chill sandwiches 15 minutes to set the filling before serving.