Sprinkle cornstarch over 1/2 cup of the nonfat milk in a small bowl and whisk to dissolve. Whisk in cocoa and sugar until smooth.

In a heavy-bottomed medium saucepan over low heat, bring the remaining 2 1/2 cups nonfat milk and the vanilla bean just to a simmer. Remove the bean, scrape the vanilla seeds into the milk, and discard the bean. Gradually whisk the cocoa mixture into the milk.

Cook, whisking often, until the mixture comes to a full boil, about 3 minutes. Remove from the heat and, if using vanilla extract, stir it in now.

Add chopped chocolate and hot half‑and‑half; mix until smooth. Strain through a wire sieve into four mugs and serve at once. (If not using immediately, cover surface with a piece of plastic wrap to prevent a skin from forming.)

Cook’s Note: Any kind of flavor (preferably sugar-free) can be added. Try raspberry, cinnamon, or holiday ginger-spice. Or, substitute milk or white chocolate for the bittersweet chocolate.