Using an electric mixer, beat the butter until soft. With the mixer on low, gradually add the confectioners' sugar until creamy, then beat in the egg. In a medium bowl, whisk together the flour, cocoa and salt; add to the mixer about one-third at a time, beating at low speed until smooth.
Lay a 16-inch sheet of parchment paper on a work surface; turn out the dough onto the paper. Using your hands, shape the dough into a 12-inch-long cylinder; wrap tightly in the parchment and roll into a smooth, even log. Twist the ends to seal and refrigerate for 30 minutes; reshape the cylinder and refrigerate for 30 minutes more.
Preheat the oven to 350 degrees ; line 2 cookie sheets with parchment paper. Unwrap the dough; using a sharp knife, slice into 1/3-inch-thick rounds. Place about 1 inch apart on the cookie sheets; bake until firm, 18 to 20 minutes. Let cool completely.
In a bowl, combine two-thirds of the chopped chocolate with the oil; microwave for 30 seconds, then stir until smooth. (If needed, heat for additional 10-second intervals.) Stir in the remaining chopped chocolate until smooth.
Using an offset spatula, spread a scant teaspoon of melted chocolate on top of each cookie. Sprinkle the peppermints on top and let stand until set, 2 to 3 hours.
The Do's and Don'ts of Cooking with Chocolate