Whisk together the flour, cocoa powder, baking soda and salt in a medium bowl.
Beat the butter in a large bowl using an electric mixer on medium-high speed until smooth. Add the sugar and beat until light and fluffy, scraping down the sides of the bowl occasionally, about 4 minutes. Beat in the egg and vanilla until smooth. Add the melted chocolate and beat until combined.
Reduce the mixer speed to medium-low and add the flour mixture in two batches, scraping down the sides of the bowl as needed. Mix until incorporated. Divide the dough in half and shape into two disks. Wrap in plastic wrap and refrigerate until firm at least 1 hour up to 3 days.
Position the oven racks in the upper and lower third of the oven and preheat to 350 degrees F. Line two baking sheets with parchment.
Working with one disk at a time, roll out to about 1/8-inch thick between two pieces of flour dusted parchment. Cut out circles with a 2-inch round biscuit or cookie cutter and transfer to the prepared baking sheets. Gather up the scraps, re-roll and cut out, refrigerating the dough if it becomes too soft.
Bake the cookies until the tops look dry and are firm, about 8 minutes. Cool for 3 to 5 minutes on the baking sheets and then transfer to a wire rack to cool completely. Repeat with the remaining dough.
When measuring flour, we spoon it into a dry measuring cup and level off the excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)
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