Gently bring the milk to a simmer in small heavy bottomed saucepan. Whisk a little of the milk into the sugar and cocoa powder and stir until no lumps remain.
Return to the pan and stir in the chocolate chips and vanilla and reheat, stirring occasionally, until the chocolate is completely melted. Turn off heat and whisk in the Baileys.
Pour the hot chocolate into each mug, then top with whipped cream. Garnish with marshmallows and more chocolate.