In a small saucepan, combine the milk, sugar, sifted cocoa powder, and cornstarch. Whisk thoroughly to dissolve the dry ingredients. Place the pan over low heat and cook the mixture, stirring constantly to prevent scorching. Allow it to slowly come to a gentle boil—this should take about 5 minutes—until the mixture thickens to a rich, pudding-like consistency.
Take the saucepan off the heat and immediately stir in the dark chocolate pieces along with a pinch of kosher salt. Continue whisking until the chocolate is completely melted and the hot chocolate becomes smooth, glossy, and deeply luscious, about 2 minutes more.
Pour the thickened hot chocolate into your favorite mugs or cups and serve right away, accompanied by crisp shortbread cookies for dipping.