Brew the espresso: Brew two shots of espresso or 1/2 cup strong coffee and divide it between two heatproof mugs.
Melt the chocolate sauce: Stir the dark chocolate sauce into the hot espresso until it turns glossy and fully dissolves, about 30 seconds.
Add mint and sugar: Whisk in the peppermint extract, vanilla extract, and sugar until no sugar grains remain at the bottom of the mug.
Heat the milk: Pour the milk into a small saucepan and heat over medium-low until steam rises and small bubbles ring the edge, about 3 minutes.
Froth the milk: Froth the warm milk with a handheld frother until it doubles in volume and holds soft peaks, about 20 seconds.
Combine coffee and milk: Pour the frothed milk slowly down the side of each mug so the foam layers on top of the chocolate-espresso mixture instead of blending in.
Top and finish: Top each mug with whipped cream and a sprinkle of crushed candy canes, then serve warm.
