Whisk cornstarch, sugar, cocoa powder, cinnamon, and salt in a medium saucepan. Add milks, whisking. Bring to a boil over medium-high heat, whisking constantly. Cook until thickened, about 1 minute. Whisk in chocolate; cook until chocolate has melted, about 1 minute.
Divide evenly among six 4-ounce cups. To prevent a skin from forming, press plastic wrap onto surface of pudding. Refrigerate until set, at least 1 hour and up to overnight. Serve garnished with chocolate shavings and cinnamon sticks.