In a pot, whisk together milk, heavy cream, sugar, vanilla and peppermint extracts, and salt. Heat on the stove on medium-low until bubbles start to form around the side of the pan.
Reduce heat to low, add chocolate and extracts, and whisk until melted, glossy, and thickened slightly, 1-2 minutes.
Pour into a mug and top with lightly-sweetened whipped cream or marshmallows and a sprinkling of crushed candy canes.
This hot chocolate is a very rich treat, much like a European drinking chocolate. A small serving is usually perfect for me. If you prefer your hot chocolate a little lighter, decrease chocolate to 2 ounces (about 1/3 cup). You can also omit the heavy cream and use 1-1/4 cups milk.
Peppermint White Hot Chocolate:
Omit the sugar and substitute 2 ounces (1/3 cup) finely chopped white chocolate or white chocolate chips for the semisweet chocolate.
Spike it for the adults:
Omit the peppermint extract and substitute 2-3 tablespoons peppermint schnapps.