Cream butter and light-brown sugar in the bowl of a mixer until light and fluffy, about 3 minutes. Add eggs and vanilla, and beat until well combined. Add cooled melted chocolate. With mixer on low speed, alternate adding dry ingredients and milk until just combined. Divide dough into quarters, wrap with plastic, and chill in the refrigerator until firm, about 2 hours.

Using a small ice-cream scoop, scoop mounds of dough, about 2 teaspoons each, onto a parchment-lined baking sheet. Chill for 30 minutes. Using your hands, roll each scoop of dough into a ball and roll in confectioners' sugar to coat heavily and completely. Place cookies 2 inches apart on a parchment-lined baking sheet. Bake until cookies have flattened and sugar splits, 11 to 13 minutes.