Lightly grease 2 baking sheets or line with parchment paper. Set aside.
In a small saucepan over low heat, combine the butter and chocolate. Cook, stir occasionally, until they are melted and the mixture is smooth.
Pour the chocolate mixture into a large bowl and let cool slightly. Stir in the granulated sugar until evenly moistened. Add the egg and vanilla, beat until light and fluffy.
Sift the flour, cocoa, baking soda, and salt together onto a sheet of waxed paper. Gradually add the flour mixture to the chocolate mixture and stir to combine. Cover and refrigerate until firm, about 1 hour.
Preheat the oven to 375 degrees. Remove the cookie dough from the refrigerator. Shape the dough into ¾” balls and roll in the powdered sugar to coat completely. Place the balls about 1 1/2” apart on the prepared sheets. Bake the cookies until puffed and cracked on top (10-12 minutes). They may appear underdone in the center but will turn crisp as they cool. Let cool on the sheets for 2-3 minutes, then transfer the cookies to a wire rack to cool completely.