In a small bowl, whisk together black cocoa and melted butter. Let cool slightly.

In the bowl of a stand mixer fitted with the paddle attachment, beat softened butter, brown sugar, and salt at medium speed until fluffy, 3 to 4 minutes, stopping to scrape sides of bowl. Beat in cocoa mixture. Add egg and vanilla, beating until well combined. Add flour, and beat until well combined and a smooth dough forms.

Divide dough in half (about 1 cup or 263 grams each). Between 2 sheets of parchment paper, roll each half to ¼-inch thickness. Freeze dough between parchment until set, about 15 minutes.

Preheat oven to 325°F (170°C). Line baking sheets with parchment paper.

Working in batches, remove top sheet of parchment. Using a 2-inch round cutter, cut dough, twisting to release from parchment. Place ½ inch apart on prepared pans. (If dough tears when removing from parchment, freeze for 5 minutes.) Reroll scraps between 2 sheets of parchment paper to ¼-inch thickness, and freeze for at least 15 minutes before cutting.

Bake until fragrant, tops are dry, and a slight indentation is left when pressed with a finger, 6 to 8 minutes. Let cool on pans for 10 minutes. Remove from pans, and let cool completely on wire racks.