Line a quarter sheet pan with parchment paper.
To make the filling: In the bowl of a stand mixer fitted with the paddle attachment (or using a hand mixer), add the butter, powdered sugar, cocoa power, and 1 tablespoon of milk.
Mix on low until combined into a cookie dough like texture.
Using your hands, form the chocolate mixture into 12 balls, each about ½-inch to ¾-inch in diameter, and place them on the prepared baking sheet. Place the baking sheet in the freezer until the balls harden, at least 15 minutes.
Using a double boiler and stirring frequently, gently melt the dark chocolate until smooth. Remove from heat. (Alternatively, you can microwave the chocolate in 30 second intervals, stirring between each one to prevent burning, until melted completely).
Remove the chocolate balls from the freezer. Lifting with a fork (do not pierce) so the excess chocolate can drip off, dip the balls into the melted chocolate and place them back on the baking sheet.
Drizzle leftover dark chocolate over the bombs.