In a medium saucepan, combine the milk, heavy cream, sugar, cocoa powder, and a pinch of salt. Heat over low heat, stirring until the sugar is completely dissolved.
Add the finely chopped dark chocolate and stir until completely melted and smooth.
In a separate bowl, whisk the egg yolks. Gradually whisk in a small amount of the hot chocolate mixture to temper the eggs.
Pour the tempered egg mixture back into the saucepan and cook over low heat, stirring constantly, until the mixture thickens slightly (about 5 minutes).
Remove from the heat, add the vanilla extract, smoked paprika, and barbecue sauce. Stir well to combine.
Let the mixture cool completely, then pour it into an ice cream maker and churn according to the manufacturer's instructions.
If you don't have an ice cream maker, pour the mixture into a shallow container and place it in the freezer. Stir every 30 minutes to prevent ice crystals from forming.
Once the ice cream is fully frozen (after about 4-5 hours of freezing), serve with chocolate shavings and a drizzle of barbecue sauce.
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