Pour the milk and heavy cream into a medium saucepan and warm over medium heat, watching carefully until small bubbles form at the edges and steam begins to rise.
Whisk in the sifted cocoa powder and sugar, stirring constantly until fully dissolved and the mixture is smooth and steaming, about 2 minutes.
Add the chocolate chips to the saucepan and stir until melted, glossy, and fully incorporated into the hot mixture.
Remove the saucepan from heat and stir in the peppermint extract, starting with a teaspoon and adding more if you prefer a stronger mint flavor.
Taste the hot chocolate and adjust sweetness or mint as desired, adding more sugar or peppermint extract if needed.
Pour the hot chocolate into mugs, then top generously with whipped cream and sprinkle with crushed candy canes for a festive touch.