Place milk, cocoa powder, sugar, chocolate chips, and cinnamon in a saucepan set over medium heat. Whisk frequently until the cocoa powder and sugar have dissolved, the chocolate chips have melted, and the mixture is hot (do not boil). Add the Bailey's and vanilla extract and stir until heated through. Transfer to mugs and serve immediately. Top with whipped cream, marshmallows, or chocolate sauce, if desired.
You could also add a splash of strong hot coffee to the hot chocolate.