Position the racks the upper and lower thirds of the oven and preheat the oven to 350°. Line the baking sheets with parchment paper.
In a medium bowl, combine the flour, baking soda, ginger, cinnamon, allspice, and salt and mix thoroughly with a whisk. Set aside.
Combine the warm butter, molasses, both sugars, fresh ginger, and the egg in a large bowl and mix thoroughly. Add the flour mixture and ginger chips and stir until incorporated. The dough will be soft.
Form the dough into 1-inch balls (1/2 ounce of dough for each). Roll balls in the Demerara sugar and place them 2 inches apart on the baking sheets. Bake, rotating the sheets from back to front and top to bottom about halfway through the baking, for 10 to 12 minutes, or until they puff up and crack on the surface and then begin to deflate in the oven. For chewier cookies, remove them from the oven when at least half or more of the cookies have begun to deflate; for crunchier edges with chewy centers, bake a minute or so longer.
Slide the parchment sheets of cookies onto cooling racks or use an offset spatula to transfer the cookies, and cool completely. Stored in an airtight container, gingersnaps keep for several days.
Excerpted from Pure Dessert by Alice Medrich (Artisan Books). Copyright 2007. Leigh Beisch photographer.