In a bowl, whisk flour, cinnamon, ginger, allspice, baking soda and salt. In a large bowl, using an electric mixer on medium-high speed, beat brown sugar and egg until smooth and light, about 5 minutes. Add applesauce and molasses and beat for 3 minutes more. Scrape down sides of bowl. On low speed, gradually beat in flour mixture until just incorporated. Transfer to a clean bowl, cover and refrigerate until firm, at least 2 hours and up to overnight.
Place racks in top and bottom thirds of oven; preheat to 350ºF. Line 4 large baking sheets with parchment.
Using a 1-inch cookie scoop, drop dough on baking sheets, 2 inches apart. (Keep remaining dough refrigerated.) Sprinkle tops with sparkling sugar. Bake, switching sheets halfway through, until just set, about 12 minutes. Let cool completely on sheets on wire racks. Slide a small spatula underneath each cookie to lift cleanly off parchment. Repeat with remaining dough and sparkling sugar.