Position racks in the upper and lower thirds of the oven and preheat to 350 degrees F. Coat 2 baking sheets with cooking spray. Whisk the flour, ginger and salt in a medium bowl. Beat the butter, brown sugar and molasses in a large bowl with a mixer on medium speed until light and fluffy, about 3 minutes. Reduce the mixer speed to low and gradually beat in the flour mixture until incorporated.
Roll out the dough until 1/4 inch thick on a lightly floured surface. Cut out cookies with 3-inch cookie cutters; arrange 1 inch apart on the prepared baking sheets. Gather the scraps and reroll to cut out more cookies. Bake, switching the pans halfway through, until the cookies are lightly browned around the edges, 14 to 16 minutes. Let cool completely on the baking sheets.
Whisk the confectioners' sugar and lemon juice in a small bowl until smooth and thick. Transfer the icing to a small resealable bag. Snip a small corner and decorate the cookies. Let set 15 minutes.