For the cookies:
In a medium bowl, whisk the flour, cinnamon, baking soda, ginger, allspice, cardamom, nutmeg and salt. In a large bowl, using an electric mixer at medium speed, beat the butter and both sugars, occasionally scraping down the bowl, until light and fluffy, about 3 minutes. Add the eggs 1 at a time, beating to blend between additions. Reduce speed to low; add the flour mixture and beat until just blended. Divide the dough in half; flatten into 2 disks. Wrap in plastic and refrigerate until firm, about 1 1/2 hours.
Position the racks in the upper and lower thirds of the oven; preheat to 350 degrees . Working with 1 dough disk at a time, roll out on a lightly floured surface to 1/4 inch thick. Cut out cookies with a 3-inch cookie cutter. Gather up the scraps; roll out again, refrigerating the dough if too soft to cut. Place the cookies 1 inch apart on 2 parchment-lined baking sheets; refrigerate 15 minutes.
Bake the cookies, rotating the sheets halfway through, until golden around the edges, 10 to 15 minutes. Transfer the cookies to racks and let cool completely. Repeat with remaining dough.
For the icing:
In a large bowl, using an electric mixer, starting at low and increasing to medium speed, beat confectioners' sugar, egg white and 3 tbsp. water until stiff, about 3 minutes. Spoon the icing into a pastry bag or a plastic freezer bag with a corner cut off. Decorate the cookies with icing; top with sprinkles or sugar, if using. Let the cookies stand until the icing sets, about 1 hour.