Prepare the Milk Mixture: Reserve ½ cup of the milk and set it aside. In a medium saucepan, combine the remaining 3 ½ cups of milk, salt, cocoa powder, and granulated sugar. Bring the mixture to a simmer over medium heat, stirring constantly to dissolve the cocoa and sugar.
Add the Dark Chocolate: Once the milk mixture is simmering, add the chopped dark chocolate. Stir continuously until the chocolate is completely melted and the mixture is smooth.
Make the Cornstarch Slurry: In the reserved ½ cup of milk, add the cornstarch and stir until it is fully dissolved, forming a slurry.
Thicken the Hot Chocolate: With the chocolate mixture still simmering, slowly pour in the cornstarch slurry while whisking or stirring constantly. Continue cooking and stirring for 1-2 minutes, until the mixture thickens to your desired consistency.
Finish and Serve: Remove the hot chocolate from the heat and stir in the vanilla extract. Serve immediately, hot and steamy.