Use metal blade to process ginger and sugar until finely chopped, about 2 minutes. Cut butter into 1" pieces. Add to work bowl and process until blended, about 30 seconds. Add molasses and egg and process until combined, about 15 seconds. Add dry ingredients and pulse to incorporate, 7 to 8 pulses. Remove dough to plastic storage bag and pat into a disc; chill until cold, at least 2 hours. Preheat oven to 350 degrees. Grease baking sheets. On a floured surface, roll out 1/4 of the dough to 1/8" thickness. Use cookie cutters to cut out desired shapes. Transfer to baking sheets and bake until firm and lightly brown, about 10 minutes.