Add milk, heavy cream, brown sugar, and cocoa powder to a pan and mix well until combined.
Cook on medium heat until the milk comes to a boil, stirring frequently to prevent scorching at the bottom of the pan.
Once the milk comes to a boil, add chocolate chips and cook until they dissolve and the hot chocolate is thick and creamy. Stir continuously while cooking.
Finally, add vanilla extract.
Add Kahlua and mix well.
Pour the hot chocolate into serving mugs.
Garnish with whipped cream, marshmallows, and a sprinkling of instant coffee powder or sprinkles, and serve immediately.