STIR 2 cups flour with ginger, cinnamon, allspice, salt and baking soda in a large bowl. Make a well in centre.
BEAT butter with brown sugar in a bowl, using an electric mixer on medium, for 2 min. Beat in egg and molasses. Pour molasses mixture into flour mixture. Stir until all flour is incorporated. Stir in remaining flour, 1/4 cup at a time, until dough is no longer sticky but still soft. Divide dough into 4 portions. Form into discs. Wrap in plastic wrap. Refrigerate until chilled, at least 30 min or up to 1 week, or freeze up to 1 month.
PREHEAT oven to 350F. Line 2 baking sheets with parchment, or spray with cooking spray. Roll out 1 disc of dough on a floured surface until 1/8 in. thick. Cut into shapes using cookie cutters. Transfer to prepared baking sheets.
BAKE in centre of oven until deep brown around edges and firm to the touch, about 7 min. Larger cut-outs may require more baking time. Repeat with remaining dough. Transfer cookies to racks to cool. Cool sheets slightly before the next batch. Store in an airtight container up to 1 month.
ROYAL ICING: Beat 3 egg whites with 1/4 tsp vinegar in a large bowl using an electric mixer, until foamy. Add 5 cups sifted icing sugar until combined. Beat on high until dense and thick, but foamy, 3 to 7 min. Stir in 2 to 4 tbsp cold water to thin the icing. Transfer to a piping bag fitted with a very small plain tip to decorate cooled cookies. (Prep tip: Let chilled dough sit for 5 min before rolling out. It will be easier to roll.)