In a standing mixer with a paddle attachment, mix the sugar, butter, molasses, and fresh ginger on low speed until just combined.
Add the ground ginger, baking soda, cinnamon, chipotle, cloves, allspice, and salt. Beat on medium speed, scraping as needed, until fluffy and pale, about 5 minutes.
Add the egg and the rosemary, and beat until smooth.
Add the flour in 3 parts, mixing on low speed, to incorporate. Chill the dough for 1 hour or up to overnight.
Meanwhile, adjust the oven rack to a medium rung, and preheat oven to 350°. Line 2 baking sheets with parchment paper. Place the turbinado sugar in a wide, shallow bowl.
Using a 1 tablespoon cookie scoop (#60), drop the dough, a dozen or so at a time, into the sugar, and using your hands, toss to coat. Arrange the dough, rounded-side up, about 2 inches apart on the baking sheets. Bake, rotating once, for 16 minutes. Transfer to a cooling rack.