Brown your butter well and place into a bowl into the fridge. Let cool in fridge until hardened. Remove from fridge about 30 minutes before you want to mix up this cookie dough.
Preheat oven to 350 degrees.
Sift the flour, ground ginger, baking soda, cinnamon, salt, and allspice together into a mixing bowl.
Mix sugar and butter with a mixer until smooth and creamy. Mix in the egg and molasses. Gradually mix in the flour mixture until just coming together into a batter.
Roll rounded teaspoon-sized amounts of dough into small balls between your hands, roll in sugar, and place onto parchment lined baking sheets.
Bake until the cookies are barely set, the tops are slightly cracked, about 10 minutes. Cool on the cookie sheet for a few minutes until they can be moved to a wire rack without breaking.