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To make the dough: In a medium bowl, sift together the flour, cocoa, cinnamon, baking soda, salt, ginger, and allspice.
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In a large bowl or the bowl of your stand mixer, beat together the butter, sugar, and vanilla on medium-high speed until the mixture is noticeably light and fluffy. Beat in the egg until completely absorbed, then the molasses.
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Reduce the mixer speed to low and stir in the dry ingredients. Mix until well blended. Scrape the bowl and mix for another 30 seconds until the dough is thoroughly combined.
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Cover and refrigerate the dough for 2 hours. When you're ready to bake, preheat the oven to 350°F. Line two baking sheets with parchment.
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To shape and bake the cookies: Pour the turbinado sugar onto a plate. Using a teaspoon scoop, portion the dough into balls. Roll the balls in the turbinado sugar, coating them completely. Place on the prepared baking sheets.
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Bake the cookies until set on the edges, but still very soft in the centers, about 10 minutes. Remove them from the oven. While still hot, make a deep indentation in the center of each cookie. Return to the oven to finish baking, about 5 minutes more. Let the cookies cool on the pans on racks.
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To make the filling: Place the chocolate and butter in a heatproof bowl. Melt in the microwave in 30-second bursts, stirring well after each interval. Stir in the molasses.
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Transfer the filling to a small zip-top bag or pastry bag and work it toward the center of the bag. Snip off a tiny bit at the corner. Fill each cookie with filling, and let sit at room temperature until the ganache is firm.
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Storage information: Store completely cooled cookies, well wrapped, at room temperature for several days; freeze for longer storage.
