Sift the flour, cocoa powder, baking soda, and salt into a large bowl; set aside.

In the bowl of a mixer fitted with the paddle attachment, cream the butter and sugar on medium-high speed until fluffy, about 3 minutes. Add the egg and vanilla and mix until combined.

Reduce the speed to low and add the flour mixture in two batches, beating until just combined after each addition. Mix in the chocolate chips.

Line 2 baking sheets with parchment paper. Scoop rounded tablespoonfuls of dough and roll into smooth round balls. Place 2 inches apart on the prepared baking sheets. Refrigerate until firm, at least 1 hour or overnight.

Preheat the oven to 350 degrees F. Bake the cookies until just set, 9 to 11 minutes. (Don't overbake!) Let cool slightly, then press the handle of a wooden spoon into the center of each cookie to make a small indentation. Remove the cookies to a rack to cool completely.

Meanwhile, melt the milk chocolate in the microwave in 30-second intervals, stirring. Spoon about 1/2 teaspoon melted chocolate into the center of each cookie and top with crushed peppermints. Let set.