Preparation Position 1 rack in top third and 1 rack in bottom third of oven; preheat to 350°F. Line 2 baking sheets with parchment paper. Whisk flour, crystallized ginger, baking soda, and 1/4 teaspoon salt in medium bowl. Using electric mixer, beat butter in large bowl until creamy and light, about 2 minutes. Gradually beat in both brown sugars. Beat on medium-high speed until creamy, about 3 minutes. Add egg, molasses, fresh ginger, ground ginger, cinnamon, and cloves. Beat to blend. Add flour mixture in 2 additions, beating on low speed just to blend between additions. Place 1/3 cup sugar in small bowl. Measure 1 tablespoon dough. Roll into ball between palms of hands, then roll in sugar in bowl to coat; place on baking sheet. Repeat with remaining cookie dough, spacing cookies 1 1/2 to 2 inches apart. Bake cookies until surfaces crack and cookies are firm around edges but still slightly soft in center, about 15 minutes. Cool completely on sheets on rack. DO AHEAD: Cookies can be made up to 4 days ahead. Store in airtight containers at room temperature. Per serving (1 cookie): 93.2 kcal calories, % calories from fat, 3.5 g fat, 2.1 g saturated fat, 14.3 mg cholesterol, 14.8 g carbohydrates, 0.2 g dietary fiber 9.1 g total sugars, 14.6 g net carbohydrates, 0.9 g proteinNutritional analysis provided by Bon Appétit