In a medium bowl, whisk together the flour, cocoa, espresso, baking powder, and salt. In a separate bowl, use an electric mixer to beat the butter and brown sugar until light and fluffy. Beat in the egg until well-combined; mix in the cooled chocolate. With the mixer on low speed, gradually add in the dry ingredients; then beat in the milk just until combined. Flatten the dough into a disk; wrap in plastic. Freeze until firm, about 45 minutes.
Preheat the oven to 350°F. Line two baking sheets with parchment paper. Shape the dough into 1-inch balls. Place the powdered sugar into a medium bowl; working in batches, roll the balls in sugar twice.
Place the balls on the prepared baking sheets, 2 inches apart. Bake, rotating sheets halfway through, until the cookies have spread and the coating is cracked, 12 to 14 minutes; the cookies will still be soft to the touch. Transfer to a wire rack to cool completely.
Notes
*Store in an airtight container, at room temperature for up to 3 days.
*You may also freeze finished cookies for use on holiday cookie platters.