Position racks in the upper and lower thirds of the oven and preheat to 350 degrees F. Lightly butter 2 baking sheets. Whisk the flour, baking soda, ground ginger and salt in a medium bowl.
Beat the butter and granulated sugar in a large bowl with a mixer on medium-high speed until light and fluffy, about 5 minutes. Beat in the molasses, egg and fresh ginger. (The mixture may look curdled.) Reduce the mixer speed to low; add the flour mixture and beat until incorporated. Stir in the crystallized ginger with a rubber spatula. The dough should be soft but not sticky; if it's sticky, refrigerate 15 minutes.
Form tablespoonfuls of dough into balls and roll in the coarse sugar. Arrange about 2 inches apart on the prepared baking sheets. Bake, switching the position of the pans halfway through, until the edges are set but the centers are still soft, 12 to 14 minutes. Let cool 5 minutes on the baking sheets, then transfer to racks to cool completely.