Melt butter in a medium sauce pan. Remove from heat and stir in brown sugar, granulated sugar, molasses, and grated ginger. Set aside to cool slightly.
Meanwhile, combine flour, baking soda, ground ginger, cinnamon, nutmeg, allspice, cardamom, cloves, and salt. Whisk well and set aside.
Add egg to the butter mixture, stirring well to combine. Add the flour mixture and stir to combine. Cover the pot and put it in the refrigerator to chill for at least an hour (the dough will be slightly sticky still after an hour; if you don't like getting your hands sticky, chill for a longer period).
Fill a small shallow bowl with coarse sugar. Using a tablespoon or small spoon, scoop out a ball of dough. Roll the dough between your palms to form a 1-inch ball. Roll the ball in the sugar, coating evenly, and place on a baking sheet lined with parchment paper. Continue with remaining dough, spacing balls about 2 inches apart.
In an oven preheated to 350° F, bake for 10 minutes, rotating the baking sheets after 5 minutes. Carefully transfer cookies to wire racks to cool.
If desired, sandwich a scoop of vanilla ice cream between two cookies before eating.