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For the cookies: In a small saucepan, melt 1/2 stick (4 tablespoons) of the butter over medium heat. Stir in the cocoa powder and mix until smooth, forming a paste. Set aside to cool. In a medium bowl, whisk together the flour, baking powder and salt.
In the bowl of a stand mixer, add the remaining 1 1/2 sticks (12 tablespoons) butter, the granulated sugar, brown sugar and cooled chocolate paste and mix until combined, about 3 minutes. Add in the vanilla and eggs and mix well. Slowly incorporate the dry ingredients into the wet ingredients and mix well.
Transfer the dough to a floured work surface, divide into two disks, wrap in plastic and chill in the refrigerator for about an hour.
Preheat the oven to 350 degrees F. Line two baking sheets with parchment paper.
Working with one disk at a time, roll out the dough to 1/4-inch thickness. Cut out stocking shapes with the cookie cutter and place the cookies 2 inches apart on the prepared baking sheets. Re-roll the scraps to get as many cookies as possible, returning the dough to the refrigerator to chill if necessary.
Bake the cookies just until they're set, 10 to 12 minutes. (Bake one sheet at a time for even cooking and browning.) Transfer the cookies to a wire rack and let cool.
For the salted caramel: In a heavy-bottomed saucepan , combine the granulated sugar and 1/4 cup water over medium-low heat and cook until the sugar dissolves. Increase the heat and bring the mixture to a boil without stirring. If necessary, use a wet pastry brush to wash down any crystals on the sides of the pan. Boil until the syrup is a deep amber color, 5 to 6 minutes.
Decorate the cookies with the chilled caramel.