Heat ½ cup of the milk and chocolate on low in a sauce pan and whisk to combine as it melts.
In separate pan on low heat, whisk the egg yolk until smooth and heated, being careful not to curdle. (excessive heat will curdle the yolk)
Slowly add chocolate mixture by the spoonful into egg yolks, while whisking continually. Add the remaining milk and peppermint extract and heat to temperature.
Add hot coffee and serve with whipped cream on top and peppermint candy cane for garnish.
Notes
Egg yolk can be omitted but I recommend using it because it adds a delectably thick and creamy mouthfeel.