STIR flour, ginger, cinnamon, allspice, baking soda and salt in a medium bowl.
BEAT butter and brown sugar in a large bowl, using an electric mixer on medium, for 2 min. Beat in egg and molasses. Gradually add flour mixture. Divide dough into 4 portions. Form into discs. Wrap in plastic wrap. Refrigerate 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. Roll out 1 disc of dough on a floured surface to ⅛-in. thickness. Cut into shapes using cookie cutters. Transfer to prepared baking sheets.
BAKE 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.
STIR egg whites, sugar, 3 tbsp water, corn syrup, cream of tartar and salt in a large metal bowl set over a large pot with 1 in. of simmering water until sugar dissolves, about 3 min. Beat with an electric mixer on medium-high over water, until stiff peaks form when beaters are lifted, about 7 min. Remove from heat.
BEAT in vanilla until combined. Use to sandwich gingerbread cookies together. Store cookies in an airtight container for up to 1 month. Tip: Allow chilled dough to sit for 5 min before rolling out. It will be easier to roll.