Warm the milk: In a small saucepan over medium heat, add milk (and half-and-half if using). Heat until steaming, not boiling.
Whisk in cocoa and sugar: Sprinkle in cocoa powder and sugar.
Whisk until smooth with no lumps.
Melt the chocolate: Add the chopped chocolate and a pinch of salt. Stir until fully melted and glossy.
Add vanilla: Remove from heat and stir in vanilla extract.
Stir in Baileys: Pour in Baileys and whisk gently. Taste and adjust sweetness if needed.
Serve: Divide into two mugs.
Top with whipped cream and chocolate shavings if you like. Enjoy warm.