01 - In a medium saucepan, whisk together milk, heavy cream if using, cocoa powder, and sugar over medium heat until smooth and steaming, avoiding boiling.
02 - Add the semisweet chocolate chips or chopped chocolate and continue whisking until fully melted and the mixture is homogeneous.
03 - Remove the saucepan from heat, then stir in peppermint and vanilla extracts thoroughly.
04 - Pour the hot cocoa into mugs and optionally top with whipped cream, crushed peppermint candies, and chocolate shavings.
05 - Serve immediately while warm.