Warm the dairy: Add milk and cream to a medium saucepan.
Heat over medium-low until steaming, not boiling. Stir occasionally to prevent a skin from forming.
Whisk in cocoa and sugar: Sift in the cocoa powder to avoid lumps. Add sugar and a pinch of salt.
Whisk until fully dissolved and smooth.
Melt the chocolate: Add chopped semi-sweet chocolate. Stir gently until it melts completely and the mixture is glossy and uniform.
Add vanilla: Stir in the vanilla extract. Taste and adjust sweetness if needed.
Remove from heat and spike: Take the pan off the burner.
Stir in Baileys. Start with 2 ounces, taste, and add more if you want a stronger flavor.
Serve right away: Pour into warm mugs. Top with whipped cream, a light cocoa dusting, or chocolate shavings for a little flair.