Pour the milk into a small saucepan and warm it over medium-low heat until it gently steams, just below boiling point, filling your kitchen with a cozy aroma.
Whisk in the cocoa powder, honey, and sugar until fully dissolved and the mixture is smooth and fragrant, about 2 minutes.
Add the fresh peppermint leaves to the saucepan and allow them to steep gently in the warm mixture for about 5 minutes, releasing their bright aroma.
Remove the saucepan from heat and strain out the peppermint leaves using a fine-mesh strainer, ensuring the hot chocolate is smooth and free of leaves.
Return the strained mixture to low heat, then add a drop of peppermint extract, stirring well to combine and intensify the mint aroma.
Froth the hot chocolate using a whisk or frother until it develops a light, velvety foam—this adds a luxurious touch and enhances the texture.
Pour the hot chocolate into mugs, garnish with a sprig of fresh peppermint if desired, and serve immediately while warm and fragrant.