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Combine the molasses and egg in the bowl of a stand mixer fitted with the paddle attachment on low speed. Add the sugar and beat on medium speed until combined. Add the melted and cooled butter and beat on low speed until thoroughly incorporated. Scrape down the sides of the bowl and paddle as needed. Add the flour mixture in thirds on low speed and mix until incorporated. Stop the mixer and scrape down the bowl and paddle between additions.
Place a sheet of plastic wrap on a work surface and place the dough in the center. Flatten the dough into a uniform thickness. Wrap with plastic wrap and place in the refrigerator to chill for 20 to 30 minutes.
Line a plate with parchment paper. Roll heaping tablespoons of the dough into balls about 1 1/2 inches in diameter and place on the plate. Refrigerate until chilled, at least 1 hour.
Position 2 racks in the top and bottom thirds of the oven and preheat the oven to 350 degrees F. Line 2 baking sheets with parchment paper.
Place a scoopful of sugar in a medium bowl for garnishing the cookies. Remove the plate from the refrigerator and, working with 6 dough balls at a time, toss them in the sugar until evenly coated. Transfer the sugared dough balls to the prepared baking sheets, evenly spacing them about 2 inches apart. Repeat with the rest of the dough balls.
Bake for 8 minutes, then rotate the baking sheets from front to back and top to bottom. Bake until the cookies are dark golden brown with lighter cracks, 7 minutes. They will be a little puffy but will deflate slightly as they cool. Slide the parchment paper with the cookies onto a wire rack to cool completely.