In a small saucepan, mix the Baileys, milk, dark chocolate, and vanilla extract. Heat gently, stirring until the chocolate has melted and the mixture is smooth.
In another saucepan, bring water, sugar, butter, and salt to a boil. Remove from heat and add flour. Stir until the mixture forms a dough.
Heat oil in a deep pan. Transfer churro dough to a piping bag and pipe strips into hot oil, frying until golden.
Mix sugar and cinnamon for coating. Roll fried churros in the cinnamon-sugar mixture.
Serve the churros warm with a mug of Baileys hot chocolate.