In a medium saucepan, whisk together the brown sugar, cocoa powder, corn starch, and salt. Add the milk and heavy cream, and whisk until combined.
Add the chocolate chips, and bring the mixture to a simmer over medium heat, stirring frequently. Simmer 2-3 minutes, or until thickened and bubbling. Remove from heat and stir in the peppermint extract.
Pour the pudding into serving dishes and cover the surface with plastic wrap to prevent a skin from forming. Refrigerate until cooled.
Serve with whipped cream and crushed candy canes on top, if desired.