In a medium bowl, whisk together the butter and granulated sugar until well combined. Whisk in the vanilla and chocolate. Add the egg and whisk until thoroughly incorporated.
In another medium bowl, stir together the flour, cocoa powder, salt, and baking soda until well mixed. Using a wooden spoon, stir the flour mixture into the chocolate mixture. The dough will start to seem too floury, and you will find it easiest to switch to mixing it with your hands until it comes together. It will have the consistency of Play-Doh. Let the dough sit at room temperature for about 1 hour to firm up.
Transfer the dough to a 15-inch square sheet of parchment or waxed paper. Using your hands, shape the dough into a rough log about 10 inches long and 2-1/2 inches in diameter. Place the log at the edge of the sheet of parchment paper, and roll the parchment around the log. With the log fully encased in parchment, roll it into a smoother log, keeping it at 2-1/2 inches in diameter. Refrigerate for at least 2 hours, or until firm. The log may settle and sink a bit in the fridge, so reroll it every 15 minutes or so to maintain a nice round log, if you like. If not, your cookies will be more oblong than round, which is not a bad thing taste-wise, though they won’t look like the famous packaged cookie. (At this point, the dough log can be well wrapped in plastic wrap and stored in the refrigerator for up to 1 week or in the freezer for up to 1 month. If the dough is frozen, thaw overnight in the refrigerator before proceeding.)
Position a rack in the center of the oven, and heat the oven to 325°F. Line a...