Heat the milk and cream: In a medium saucepan, combine 3 cups of milk and 1 cup of heavy cream. Place over medium-low heat and warm until hot but not boiling, stirring occasionally to prevent sticking.
Add white chocolate: Once the milk mixture is hot, add 1 cup of white chocolate chips into the saucepan. Stir continuously until the chocolate is fully melted and the mixture is smooth.
Incorporate the extracts: Add 1/2 teaspoon of vanilla extract and 1/2 teaspoon of peppermint extract (if using) to the hot chocolate mixture. Mix thoroughly.
Color the drink: Add a few drops of green food coloring to achieve the desired festive shade. Stir until the color is evenly distributed.
Serve the hot chocolate: Carefully pour the hot chocolate into mugs. Top each serving with a generous dollop of whipped cream, along with mini marshmallows, red & green sprinkles, and, if desired, a sprinkle of crushed candy canes for extra flavor.