In a small saucepan, simmer the stout over medium heat until reduced by about half, concentrating the flavor. This should take about 5-10 minutes. Let cool completely.
In another saucepan, heat the heavy cream and milk over medium heat until steaming, but do not boil.
Whisk in the sugar, cocoa powder, and a pinch of salt into the milk-cream mixture. Add the finely chopped dark chocolate. Stir until completely melted and smooth.
In a separate bowl, whisk the egg yolks. While whisking constantly, slowly drizzle a small amount of the hot chocolate mixture into the egg yolks to temper them.
Pour the tempered egg mixture back into the saucepan. Cook over low heat, stirring constantly, until the mixture thickens slightly and coats the back of a spoon (about 5 minutes).
Remove from heat, stir in the vanilla extract and the reduced stout beer. Let cool completely.
Pour the mixture into an ice cream maker and churn according to the manufacturer's instructions until creamy.
If you don't have an ice cream maker, pour the mixture into a shallow container and freeze, stirring every 30 minutes to prevent ice crystals from forming.
Once the ice cream is completely frozen (after about 4-5 hours of freezing), serve with chocolate shavings or extra beer chocolate sauce.
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