In a medium saucepan, whisk together sugar, cornstarch, cocoa powder and salt. Turn pan on medium heat, and add in milk, whisking to incorporate. Continue cooking until mixture is thickened and there are no clumps, around ten minutes.
Add in chocolate and mix for another 3-4 minutes.
Remove from heat and add vanilla and peppermint extract.
Give one final mix and then pour into your serving containers. Refridgerate for 30 minutes or longer. Serve garnished with peppermint chunks and/or chocolate shavings