In a food processor (or blender), whirl chopped ginger and 1/3 cup sugar until ginger is finely ground. Pour from container.
In the same container, whirl butter and 1/3 cup sugar until fluffy.
Add ginger mixture, molasses, and egg; whirl to mix.
In a bowl, mix flour, baking soda, ground ginger, cinnamon, and nutmeg. Whirl into butter mixture.
Cover dough and chill until firm to the touch, about 1 hour.
Preheat oven to 350°. Shape dough into 1-in. balls and coat in remaining sugar (use a little more sugar if needed). Place balls 2 to 3 in. apart on nonstick or oiled baking sheets.
Bake until slightly darker brown, 11 to 12 minutes, switching positions of baking sheets halfway through. Transfer cookies to racks to cool.
Make ahead: Up to 1 week, stored airtight at room temperature; frozen, up to 4 months.