Sift flour, ginger, cinnamon and baking soda together. Set aside.
Beat butter and sugar until fluffy. Beat in egg and molasses, blending well. Stir in flour mixture, 1/3 at a time, until well blended.
Roll dough into 1 tsp balls. Coat in sugar. Place on cookie sheet and flatten slightly. Bake at 350 for 9 or so minutes until tops are cracked. Cool completely.
With mixer at medium speed, beat the butter and lime zest in a medium bowl until creamy. With mixer at low speed, gradually beat in the confectioners’ sugar and enough of the lime juice to make a thick, spreadable frosting.
Set half of cookies on flat surface, outside down. Using teaspoon sized scoop or your hands, make rounded frosting ball and place onto cookie. Top with another cookie. Chill.